Power system restoration incorporating diverse distributed energy resources

ABSTRACT

An example system includes an aggregator configured to receive a service collaboration request and iteratively determine, based on minimum and maximum power values for DERs under its management, an optimized operation schedule. The aggregator may also be configured to iteratively determine, based on the optimized operation schedule, an estimated flexibility range for devices under its management and output an indication thereof. The system may also include a power management unit (PMU) configured to iteratively receive the indication and determine, based on a network model that includes the estimated flexibility range, a reconfiguration plan and an overall optimized operation schedule for the network. The PMU may also be configured to iteratively cause reconfiguration of the network based on the plan. The PMU and aggregator may also be configured to iteratively, at a fast timescale, cause energy resources under their management to modify operation based on the overall optimized operation schedule.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/983,242, titled “GRID EDGE FLEXIBILITY QUANTIFICATION ANDOPTIMIZATION” and filed Feb. 28, 2020, the entire content of which isincorporated herein by reference.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under Contract No.DE-AC36-08G028308 awarded by the Department of Energy. The governmenthas certain rights in the invention.

BACKGROUND

As renewable energy becomes more important in today's society, powergrids may have to manage increasingly distributed energy sources,including renewable energy sources. For instance, even modest housingmay have photovoltaic (PV) systems and/or wind turbines installed toreduce dependence on the grid and/or to offset energy costs. Inaddition, availability and utilization of grid-friendlier devices, suchas controllable loads and/or energy storage systems also continue toincrease. As prevalence of these distributed energy resources increases,grid managers, such as those who manage power distribution networks,will be faced with power failures and outages on more complex powernetworks.

SUMMARY

In one example, a system includes an aggregation unit comprising atleast one processor, the aggregation unit being configured to receive aservice collaboration request indicating a problem in a powerdistribution network and, responsive to receiving the servicecollaboration request, determine, based on respective minimum andmaximum real and reactive power values for one or more devices in thepower distribution network under management by the aggregation unit, anoptimized operation schedule covering a model predictive control (MPC)horizon duration for the one or more DERs. The aggregation unit may befurther configured to determine, based on the optimized operationschedule, an estimated flexibility range for DERs under management bythe aggregation unit and output an indication of the estimatedflexibility range. The system may further include a power managementunit comprising at least one processor, the power management unit beingconfigured to receive the indication of the estimated flexibility rangeand determine, based on a linearized restoration model of thethree-phase unbalanced power distribution network that includes theindication of the estimated flexibility range, an optimized powerdistribution network reconfiguration plan and an overall optimizedoperation schedule covering the MPC horizon duration for both energyresources under management by the power management unit and the one ormore DERs. The power management unit may be further configured to cause,based on the optimized power distribution network reconfiguration plan,a reconfiguration of the power distribution network and output anindication of the overall optimized operation schedule. The powermanagement unit may be further configured to cause one or more of theenergy resources under management by the power management unit to modifyoperation based on the overall optimized operation schedule. Theaggregation unit may be further configured to receive the indication ofthe overall optimized operation schedule, determine, based on theindication of the overall optimized operation schedule, setpoints forthe one or more DERs, and cause at least one of the one or more DERs tomodify operation based on the setpoints.

In another example, an aggregation unit includes at least one processorconfigured to receive a service collaboration request indicating aproblem in a power distribution network and, responsive to receiving theservice collaboration request, determine, based on respective minimumand maximum real and reactive power values for one or more DERs in thepower distribution network under management of the aggregation unit, anoptimized operation schedule covering a model predictive control (MPC)horizon duration for the one or more DERS. The at least one processormay be further configured to determine, based on the optimized operationschedule, an estimated flexibility range for devices under management bythe aggregation unit and output, for use by a management device in thepower distribution network, an indication of the estimated flexibilityrange. The at least one processor may be further configured to receivean indication of an overall optimized operation schedule covering theMPC horizon duration, determine, based on the indication of the overalloptimized operation schedule, setpoints for the one or more DERs, andcause at least one of the one or more DERs to modify operation based onthe setpoints.

In another example, a power management unit includes at least oneprocessor configured to receive an indication of an estimatedflexibility range of one or more devices in a power distribution networkthat are under management by an aggregation unit and determine, based ona linearized restoration model of the power distribution network thatincludes the indication of the estimated flexibility range, an optimizedpower distribution network reconfiguration plan and an overall optimizedoperation schedule covering the MPC horizon duration for both energyresources under management by the power management unit and one or moredistributed energy resources (DERs) under management by the aggregationunit. The at least one processor may be further configured to cause,based on the optimized power distribution network reconfiguration plan,a reconfiguration of the power distribution network and output, for useby the aggregation unit, an indication of the overall optimizedoperation schedule. The at least one processor may be further configuredto cause one or more of the energy resources under management by thepower management unit to modify operation based on the overall optimizedoperation schedule.

The details of one or more examples are set forth in the accompanyingdrawings and the description below. Other features, objects, andadvantages will be apparent from the description and drawings, and fromthe claims.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a conceptual diagram illustrating an example powerdistribution system configured to perform power system restorationincorporating diverse distributed energy resources, in accordance withone or more aspects of the present disclosure.

FIG. 2 is a flow diagram illustrating an example collaboration frameworkfor power system restoration incorporating diverse distributed energyresources, in accordance with one or more aspects of the presentdisclosure.

FIG. 3 is a conceptual diagram illustrating example timing andsequential control actions for power restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure.

FIG. 4 is a flow diagram illustrating example operations for networkreconfiguration during power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure.

FIG. 5 is a conceptual diagram illustrating an example simplified powernetwork that may employ power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure.

FIG. 6 is a graphical diagram illustrating a modified IEEE 123-bus testfeeder that was employed for testing power system restorationincorporating diverse distributed energy resources, in accordance withone or more aspects of the present disclosure.

FIGS. 7A and 7B are conceptual diagrams illustrating switch operationsduring two test cases of power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure.

FIGS. 8A-8L are graphical plots illustrating detailed results of a testcase of power system restoration incorporating diverse distributedenergy resources, in accordance with one or more aspects of the presentdisclosure.

FIG. 9 is a flow diagram illustrating example operations for performingpower system restoration incorporating diverse distributed energyresources, in accordance with one or more aspects of the presentdisclosure.

DETAILED DESCRIPTION

The present disclosure provides systems, devices, and methods for powersystem restoration incorporating grid edge flexibility. As furtherdescribed herein, the power generation capability of utility-scaledistributed energy resources (DERs) and the flexibility of dispersedDERs, including behind-the-meter DERs, are coordinated through ahierarchical structure to recover distribution system power suppliesafter outages. As one example, a power management unit (PMU), such asthat operated by a distribution system operator (DSO), may determinethat an outage has occurred and request flexibility information fromaggregators in the distribution system. The aggregators may optimizeperformance of DERs under their control and generate an estimate of DERflexibility. The PMU may receive estimates of DER flexibility and usemodel predictive control (MPC) in a restoration planning stage todynamically adjust system restoration strategies and the reference ofDERs based on up-to-date forecast data. With consideration of thethree-phase unbalanced distribution power flow, the PMU may ensureradiality of islands via various methods, such as mathematicalprogramming or a computationally efficient heuristic algorithm. The PMUmay also send operating setpoints that represent the optimal MPCsolution. The aggregators may perform real-time balancing using dispatchmodels that mitigate unexpected errors and fluctuations. In other words,DSOs and aggregators/end-users will be coordinating during the outageperiod to fully use edge DER capabilities for service restoration.

The supply and delivery of reliable electric power is the most criticaltarget of the entire power industry. Although numerous efforts have beenmade to improve power system reliability and resilience, power failuresand outages remain inevitable because of various factors including thefragility of infrastructure and the increasing threats of extremeweather and malicious attacks. Power system restoration strategies aimto minimize the negative impacts of such service interruptions onelectricity customers.

Related-art system restoration studies in the literature have emphasizedthe recovery of transmission systems, such as the black-start process,generator cranking, energizing the backbone transmission network, andisland sectionalization. Service restoration in distribution systems wasgenerally limited to network reconfiguration. With the ever-increasingintegration of distributed energy resources (DERs), modern distributionsystems may have enough generation capacity to perform proactiverestoration actions after outages, but there still exist several keyobstacles that limit the contributions of DERs in distribution systemrestoration.

By solidifying the role of smaller-scale DERs, includingbehind-the-meter resources, and providing efficient communication andcoordination of utility operators, aggregators, and end-use customers,the techniques of the present disclosure allow utility operators toleverage and rely on smaller-scale DERs to assist in distribution systemrestoration. In this way, utility operators no longer have to rely onlyon the dispatch of utility-scale DERs (e.g., distributed generators andenergy storage systems (ESSs)) with large capacities. The techniques ofthe present disclosure also address real-time balancing and uncertaintyregarding generation capability of such DERs as well as consumptionpatterns of consumers by creating a novel, collaborative framework thatintegrates restoration planning and real-time balancing and resourcedispatch.

FIG. 1 is a conceptual diagram illustrating an example powerdistribution system (e.g., system 2) configured to perform power systemrestoration incorporating grid edge flexibility, in accordance with oneor more aspects of the present disclosure. In the example of FIG. 1,system 2 includes power management unit 4 and aggregators 10A and 10B(collectively “aggregators 10”). System 2 also includes systemconfiguration devices 6A, 6B, and 6C (collectively “system configurationdevices 6”), connection point 7, aggregator-managed DERs 8A-8D(collectively “aggregator-managed DERs 8”), and PMU-managed DER 9.System 2 may include additional components that are not shown. As shownin the example of FIG. 1, system configuration devices 6, connectionpoint 7, aggregator-managed DERs 8, PMU-managed DER 9, and aggregators10 (and possibly other components) are all connected via a network ofpower lines and, with those power lines, may represent a typical systemfor distributing power to customers.

System 2, as shown in the example of FIG. 1, represents a simplifiedpower distribution system. In other examples, the power system mayinclude any number of system configuration devices, PMU-managed DERs,aggregators, and/or aggregator-managed DERs. Thus, while shown in theexample of FIG. 1 as having three system configuration devices, fouraggregator-managed DERs, one PMU-managed DER, and two aggregators, powerdistribution systems may, in other examples, have more or fewer of theseitems. Additionally, system 2 represents only one example of a systemconfigured to perform the techniques described herein, and various othersystems, having additional components, fewer components, and/or othercomponents may be used in accordance with the techniques of the presentdisclosure.

Power management unit 4, as shown in the example of FIG. 1, is a systemor device configured to manage system 2. Power management unit 4 may bea computing device, such as a server computer, a desktop computer, orany other device capable of implementing some or all of the techniquesdescribed herein. In some examples, power management unit 4 mayrepresent a cloud computing environment. That is, while shown as asingle box in the example of FIG. 1, power management unit 4 may, insome examples, be a group of distributed computing resources thatcommunicate with one another to perform at least some of the techniquesdescribed herein. In some examples, power management unit 4 may be thesame as or be physically collocated with connection point 7. In otherwords, in some examples, connection point 7 may be integrated with powermanagement unit 4 or otherwise configured to perform the operations ofpower management unit 4 as described herein. In some examples, such asthe example shown in FIG. 1, connection point 7 and power managementunit 4 may be separate from one another.

In the example of FIG. 1, system configuration devices 6 are devicesoperable to reconfigure power distribution system 2 to manage powerflow, address outages or faults, or for other reasons. For example,system configuration devices 6 may include switches (also known asswitch gear) that can connect or disconnect portions of powerdistribution system 2 to one another. In some examples, systemconfiguration devices 6 may be directly operable by power managementunit 4. In some examples, system configuration devices 6 may be operableby other devices, may be physically operable by individuals, and/or mayoperate on their own. Additional examples of system configurationdevices 6 include circuit breakers, reclosers, relays, and others.

Connection point 7, in the example of FIG. 1, is the point at whichsystem 2 is connected to a power transmission system. That is,connection point 7 is a power substation. In some examples, connectionpoint 7 may represent a connection between system 2 and a larger powerdistribution network. In other words, in some examples the techniquesdisclosed herein may be used in a subset of an entire power distributionnetwork and in some examples, they may be used in the entire powerdistribution network. As discussed above, connection point 7 may, insome examples, be collocated with and/or integrated with powermanagement unit 4.

In the example of FIG. 1, aggregator-managed DERs 8 are system-connecteddevices having controllable power output and/or consumption.Specifically, aggregator-managed DERs 8A and 8B are PV systems thatutilize sunlight to generate power. Aggregator-managed DER 8C is abattery energy storage system (BESS). Aggregator-managed DER 8D is awater heater. Additional examples of aggregator-managed DERs may includewind turbines, generators (e.g., diesel generators), heating,ventilation, and/or air conditioning (HVAC) systems, electricvehicles/electric vehicle chargers, washing machines, smartplug-connected loads, refrigeration devices, or any other electricaldevices whose generation and/or consumption of energy may be controlled.In the example of FIG. 1, the generation and/or consumption of energy byaggregator-managed DERs 8 is managed by aggregators 10 as furtherdescribed below.

PMU-managed DER 9, in the example of FIG. 1, is a large-scale PV system.In general, PMU-managed DERs may be any system-connected device orsystem having controllable power output and/or consumption. Additionalexamples of PMU-managed DERs may include wind farms, generators (e.g.,natural gas, coal, etc.), energy storage systems, and other energyresources owned or operated by utilities. In various examples, theconsumption or generation of energy by PMU-managed DERs may be directlyor indirectly managed by power management unit 4.

In the example of FIG. 1, aggregators 10 are systems or devicesconfigured to manage aggregations of energy consumption and/orgeneration devices. Specifically, aggregator 10A managesaggregator-managed DER 8A and aggregator 10B manages aggregator-managedDERs 8B, 8C, and 8D. While not shown in the example of FIG. 1, thedevices managed by aggregators 10 may include uncontrollable devices,including uncontrollable loads like lighting and miscellaneouselectrical loads. Examples of aggregators 10 may include home energymanagement systems (HEMSs) that typically manage devices in aresidential home, building energy management systems that typicallymanage devices in a commercial or industrial building, microgridcontrollers that manage devices on a campus, such as a military base, aschool, a business park, or other location, and control units ofthird-party distribution system aggregators that manage devices ofmultiple end users.

While shown in the example of FIG. 1 as connecting aggregator-managedDERs 8 to system 2, aggregators 10 may, in some examples, be separatefrom the power distribution system. That is, in some examplesaggregator-managed DERs 8 may be connected to system 2 on their own. Insuch examples, aggregators 10 may be communicatively coupled withaggregator-managed DERs 8 and/or other devices in the power distributionsystem in order to perform at least some of the techniques describedherein.

Some aggregator-managed DERs may be directly connected to the powerdistribution system and thus their operation may be more apparent tosystem operators. For instance, when a distribution system aggregator ismanaging DERs of multiple individuals, some such DERs may be directlyconnected to the power distribution network. Other aggregator-managedDERs may not be directly connected, and may instead be “behind themeter”, rendering them less apparent to system operators. For instance,when a HEMS is managing DERs of a single residential customer, a rooftopPV system, a water heater, an HVAC, and a BESS may be behind the meterand thus their specific operation may not be apparent to systemoperators.

In accordance with the techniques of the present disclosure, anaggregation unit may include one or more processors. The processors maybe configured to receive a service collaboration request indicating aproblem in a power distribution network. In the example of FIG. 1, forinstance, one or more processors (not shown) of aggregators 10 mayreceive request 11 output by power management unit 4. Such requests maybe received when power management unit 4 determines that an outage hasoccurred in system 2 or that some other problem with system 2 maybenefit from assistance of aggregators 10. In some examples, the requestmay only indicate that collaboration from the aggregators is requested.That is, the request may be rather rudimentary. In some examples, therequest may provide additional or other information for use by theaggregators.

Responsive to receiving the service collaboration request, theprocessors of the aggregation unit may determine, based on respectiveminimum and maximum real and reactive power values for one or more DERsin the power distribution network under management by the aggregationunit, an optimized operation schedule covering a model predictivecontrol (MPC) horizon duration for the one or more DERs. For instance,aggregator 10A may utilize minimum and maximum real and reactive powervalues for aggregator-managed DER 8A to determine an optimized operationschedule as described in detail below. The optimized operation schedulemay represent a minimization of power consumption by devices under themanagement of aggregator 10A while still meeting the goals (e.g.,comfort, safety, minimum or maximum power needs, etc.) of aggregator10A.

Similarly, aggregator 10B may utilize the minimum and maximum real andreactive power values for aggregator-managed DERs 8B, 8C, and 8D todetermine its own optimized operation schedule. As one example, ifaggregator-managed DER 8D (a water heater) can defer activation for awhile—such as because residents aren't using hot water or because itrecently reheated its water—then aggregator 10B's optimized operationschedule may include aggregator-managed DER 8D not using any energyduring the MPC horizon duration.

The processors of the aggregation unit may determine, based on theoptimized operation schedule, an estimated flexibility range for devicesunder management by the aggregation unit. For example, aggregator 10 amay predict the minimum amount and maximum amount of energy that wouldbe consumed or produced by aggregator-managed DER 8A and other manageddevices (not shown) if its determined optimized operation schedule werefollowed. Aggregator 10 a may also predict the minimum and maximumamount of power that would be consumed or produced by such devices ifthe optimized operation schedule were followed. In some examples,determining the estimated flexibility range may be adding up the minimumand maximum energy consumption and power consumption of each deviceunder management by an aggregator at each point in time in the MPChorizon duration. The estimated flexibility ranges thus represent howmuch devices under the aggregator's management can assist in addressinga problem within the power distribution network. In the example of FIG.1, for instance, aggregator 10B may add up the predicted minimum energyusage of each of aggregator-managed DERs 8B, 8C, and 8D at eachtimeframe in the MPC horizon duration to determine an aggregatepredicted minimum energy usage. Aggregator 10B may do the same formaximum energy usage, minimum power usage, and maximum power usage.

The processors of the aggregation unit may output an indication of theestimated flexibility range. For example, aggregators 10A and 10B mayoutput estimated flexibility ranges 12A and 12B, respectively. Estimatedflexibility ranges may be communicated using one or more wired orwireless networks (not shown). In some examples, aggregators 10 mayadditionally output an indication of their optimized operation schedule.

In accordance with the techniques described herein, a power managementunit may include one or more processors. The processors of the powermanagement unit may be configured to receive the indication of theestimated flexibility range. For instance, one or more processors (notshown) of power management unit 4 may receive estimated flexibilityranges 12A and 12B.

The processors of the power management unit may be configured todetermine, based on a linearized restoration model of the three-phaseunbalanced power distribution network that includes the indication ofthe estimated flexibility range, an optimized power distribution networkreconfiguration plan and an overall optimized operation schedulecovering the MPC horizon duration for both energy resources undermanagement by the power management unit and the one or more DERs. In theexample of FIG. 1, for instance, power management unit 4 may incorporateestimated flexibility ranges 12A and 12B into a model of system 2 anduse the model to determine how best to reconfigure system 2 and anoptimized operation schedule for operation of system 2. Further detailsregarding the model are provided below.

The processors of the power management may be further configured tocause, based on the optimized power distribution network reconfigurationplan, a reconfiguration of the power distribution network. For instance,in the example of FIG. 1, power management unit 4 may output switchinstruction 13 to cause system configuration device 6B (e.g., a switch)to open, thus reconfiguring system 2. Reconfiguration may be used toensure proper system architecture for addressing issues in the powerdistribution network.

The processors of the power management unit may also output anindication of the overall optimized operation schedule. For instance,power management unit 4 may output optimized schedule 14 for use byaggregators 10.

The processors of the aggregation unit may be further configured toreceive the indication of the overall optimized operation schedule anddetermine, based on the indication of the overall optimized operationschedule, setpoints for the one or more DERs. For instance, aggregators10 may each receive optimized schedule 14 and determine setpoints fortheir respective ones of aggregator-managed DERs 8. Aggregators 10 maydetermine setpoints based on various criteria, including interests ofthe aggregator (e.g., comfort) and limitations of the DERs undermanagement (e.g., maximum power) as further described herein.

The processors of the aggregation unit may cause the one or more DERs tomodify operation based on the setpoints. For example, aggregator 10A maycause aggregator-managed DER 8A to modify operation and aggregator 10Bmay cause one or more of aggregator-managed DERs 8B, 8C, and 8D tomodify operation.

The processors of the power management unit may cause one or more of theenergy resources under management by the power management unit to modifyoperation based on the overall optimized operation schedule. Forexample, power management unit 4 may generate setpoints for PMU-managedDER 9 and output an indication as setpoints 15. This may causePMU-managed DER 9 to modify operation.

As further described herein, components of system 2 (e.g., powermanagement unit 4, aggregators 10) may be configured to perform thetechniques described herein in an iterative fashion that allows system 2to ensure maximum coordination in real-time operation while coping withthe computational complexity of model predictive control. In particular,once aggregators 10 receive the service collaboration request,aggregators 10 may determine their optimized operation schedule,determine their estimated flexibility ranges, and output 12A and 12Biteratively at a first frequency that represents an MPC interval.Correspondingly, power management unit 4 may determine the optimizedpower distribution network reconfiguration plan and overall optimizedoperation schedule, cause the reconfiguration of the power network, andoutput the overall optimized operation schedule at the MPC intervalfrequency. The MPC interval may be 5 minutes, 15 minutes, 30 minutes, orother appropriate duration. At this frequency, the power management unitand aggregation units may be sufficiently capable of performing themodeling necessary to adequately predict how the power distributionsystem will operate. In this way, the power management unit andaggregation units may regularly re-determine the predicted optimumsolutions for collaboratively addressing any issues within the powerdistribution network.

However, no prediction will be perfect in reality, and network variablesmay fluctuate from minute to minute or more. Thus, the power managementunit may cause the PMU-managed DERs to modify operation iteratively at asecond, higher frequency that represents a real-time interval.Similarly, the aggregation units may determine setpoints for theiraggregator-managed DERs and cause their aggregator-managed DERs tomodify operation iteratively at the real-time interval frequency. Thereal-time interval may be 10 seconds, 30 seconds, 1 minute, 2 minutes,or other appropriate duration. By modifying operation of network DERs ona faster timescale, the techniques provided herein allow for efficientuse of computing resources while also allowing system operators toleverage diverse DERs in the network to address outages and otherdisturbances.

The following mathematical development and additional details anddescription are discussed with reference to a home energy managementsystem (HEMS), which may be one example of an aggregation unit asdescribed herein. It should be understood that other aggregation units,such as building energy management systems that control the energyschedules for commercial/industrial buildings, third-party aggregatorsthat aggregate the demand response of retail customers, and microgridcontrollers that manage different distributed energy resources inside acampus may also be used in accordance with the techniques describedherein. The following nomenclature is used below to describe theadditional details and description of the disclosed techniques.

Acronyms/Initialisms

DER Distributed energy resource

DSO Distribution system operator

ESS Energy storage system

HEMS Home energy management system

MPC Model predictive control

PV Photovoltaic

Sets and Indexes

Ω_(A) _(h) /a Set/index of DER at household h

Ω_(B) _(h) /b Set/index of comfort parameters at household h

ω_(CG)/n Set/index of conventional generators

Ω_(H)/h Set/index of households

Ω_(L)/nm Set/index of distribution lines

Ω_(N)/n Set/index of distribution system buses

Ω_(RG)/n Set/index of renewable generators

Ψ_(n)/ϕ Set/index of phases at bus n

τ, ω Indexes of MPC and real-time intervals

Parameters

c_(h,τ) ^(b), c_(h,ω) ^(b) External influencing factor for householdcomfort parameter

e_(h,τ) Household cumulative energy consumption

n_(t) Number of real-time intervals in one MPC time interval

P_(n,ω) ^(act) Real-time renewable generation capability

P_(n,ω) ^(G,reg) Real-time regulation capacity

r_(nm) ^(ϕϕ), x_(nm) ^(ϕϕ) Resistance and reactance of distribution line

T_(st) T_(en) Start and end time of the MPC horizon

β_(h) ^(b) Weighting coefficient of comfort preference

β_(h) ^(P), β_(h) ^(e) Penalty coefficients of real-time household powerand energy violations

δ_(h) ^(a) DER conversion efficiency factor

δ_(h) ^(C), δ_(h) ^(D) ESS charging and discharging efficiencies

γ_(n) ^(C) Renewable curtailment penalty

γ_(n) ^(ES) ESS dispatch penalty

γ_(n) ^(G) DER generating cost

γ_(n) ^(S) Penalty of load shedding

γ_(h) ^(V) Penalty of household flexibility violation

γ_(h) ^(W) Penalty of household desired power deviation

η Acceptable real-time power deviation

ω_(h) ^(a,on), ω_(h) ^(a,off) Minimum on and off time of householdappliances

ΔT, Δt Length of MPC and real-time intervals

Variables

c_(h,ω) ^(P), c_(h,ω) ^(e) Real-time penalty for HEMS power schedule andenergy schedule violation

p_(h,τ) ^(a), p_(h,ω) ^(a) Active power load of household DER

P_(h,τ), Q_(h,τ) Household loads optimized by DSO

P_(n,ω) ^(curt) Real-time renewable generation curtailment

P_(n,ω) ^(ES,C), P_(n,ω) ^(ES,D) ESS charging and discharging power

P_(n,τ) ^(D,ϕ), P_(n,ω) ^(D,ϕ), Nodal active load consumption

P_(n,τ) ^(D,fix,ϕ), P_(n,ω) ^(D,fix,ϕ) Active power of theuncontrollable load

P_(n,τ) ^(G), P_(n,ω) ^(G) Active power generation of utility-scale DERs

P_(n,τ) ^(S,ϕ) Active load shedding

P_(h,τ) ^(V) Violation of household's flexibility range

P_(h,ω) ^(W) Deviation of household's preferred schedule

P_(nm,τ) ^(ϕ), Q_(nm,τ) ^(ϕ) Active and reactive line power flows

q_(h,τ) ^(a), q_(h,ω) ^(a) Reactive power load of household DER

Q_(n,τ) ^(D,ϕ), Q_(n,ω) ^(D,ϕ) Nodal reactive load consumption

Q_(n,τ) ^(D,fix,ϕ), Q_(n,ω) ^(D,fix,ϕ) Reactive power of theuncontrollable load

Q_(n,τ) ^(G), Q_(n,ω) ^(G) Reactive power output of utility-scale DERs

Q_(n,τ) ^(S,ϕ), Q_(n,ω) ^(S,ϕ) Reactive load shedding

s_(h,τ) ^(b), s_(h,ω) ^(b), Generalized comfort indicator of householdDER

ρ_(n,ω) ^(ES) ESS energy storage level

u_(hmω) ^(a) Binary control signal of household DER

V_(n,τ) ^(ϕ) Squared nodal voltage magnitude

Superscripts

min/max Minimum or maximum limit of a quantity

pre The preferred value of a quantity

The details provided below can be summarized as: (A) The framework tocoordinate DSOs and individual customers (i.e., aggregators) isintroduced based on a hierarchical structure. Post-event servicerestoration planning and real-time dispatch are integrated into theframework to fully use DER capabilities in uncertainty management. (B)Adaptive restoration planning models are provided for DSOs andindividual customers to integrate the operations of switches, thescheduling of utility-scale DERs, and the flexible responses ofbehind-the-meter DERs based on MPC techniques. A linearized three-phaseunbalanced distribution power flow is provided and, in some examples, amathematical programming approach is used to solve the distributionsystem reconfiguration problem. In some examples, a heuristicreconfiguration algorithm is employed to solve the distribution systemreconfiguration to improve computational efficiency. And (C) Withrespect to the MPC-based restoration planning solutions, real-timedispatch models are provided to address potential forecast errors in anMPC time interval and maintain real-time power balance. The DERswitching frequency is also considered in the real-time dispatch toaccount for the practical operating protocols of certain aggregatorDERs, such as behind-the-meter DERs.

As an overview, the main goal of distribution system restoration is tomaintain power supply to critical customers and minimize serviceinterruptions. To achieve this, the DSOs are supposed to reconnect theoutage area through switch operations and use available resources torestore power supply. In the past, DSOs have had access only toswitchgear and utility-owned DERs, whereas DERs are managed andcontrolled by aggregators that control various smaller-scale DERs or, insome examples, individual customers that control behind-the-meter DERs.

To utilize the flexibility of aggregator DERs for better servicerestoration performance, the techniques of the present disclosureinclude a collaboration framework to coordinate DSO's restorationdecisions with the behaviors of aggregator DERs. The uncertainty andintermittency of DERs also pose great challenges to the security andreliability of power supply during service restoration. In accordancewith the techniques described here, however, MPC-based restorationplanning and real-time dispatch are integrated to respectively handlelong-term forecast errors (e.g., hours ahead) and short-termfluctuations (e.g., minute or second level). The uncertainties of DERsmay also significantly influence the sustainability and reliability ofload supply. To achieve more sustainability, a combination of diversepower sources including renewables, energy storage, and dieselgenerators is recommended.

Note that no specific assumptions were made on the composition ofresources in developing the restoration framework and the optimizationmodels disclosed herein. The techniques of the present disclosure can beapplied to universal distribution system restoration cases.

Many DERs in power distribution networks are not owned by a DSO. Forexample, behind-the-meter DERs are mainly owned by private customers.Therefore, a DSO cannot expect the behind-the-meter DERs to cooperatewithout appropriate collaboration schemes and mutual agreement. Thetechniques disclosed herein were developed based upon the followingassumptions to enable the collaboration between DSO and thebehind-the-meter DERs during the distribution system restoration stage:(1) The behind-the-meter DERs of residential customers are the focus ofthis portion of the disclosure. However, the techniques of the presentdisclosure may be applied to DERs of commercial and industrial customersand other distribution network aggregators. (2) Each residential houseis equipped with a home energy management system (HEMS) or an equivalentgrid-edge controller with similar functionality that controls thebehind-the-meter DERs. If a house does not have any behind-the-meterDERs, there is no need to introduce a HEMS, and the corresponding housewill be treated as an uncontrollable load. (3) The distribution systemhas a hierarchical control framework where behind-the-meter DERs arecontrolled by HEMSs, and multiple HEMSs are coordinated by thesystem-level controller (i.e., DSO). It should be noted, however, thatthe techniques described herein may be trivially modified to work withadditional levels of hierarchical control in the distribution system.(4) The collaboration between the DSO and HEMSs is achieved throughdemand response programs. The DSO provides demand response programoptions that incent the enrollment of residential customers. Theseprograms will specify the responsibility of the customer (e.g., how fastshould the customer responds to a request). In return, the customer mayget credits or other monetary rewards for subscribing to a program. Itshould be noted, however, that collaboration between the DSO andaggregators may be incentivized/achieved in various other ways. (5) Thedemand response programs are typically long-term programs withfixed-term payments. Hence, any rewards are not addressed in thedisclosed techniques because they are constant parameters. On the otherhand, if a customer fails to provide a requested service as described inthe subscribed program, financial penalties specified in the program mayapply. These penalties are performance-based and are included in thedisclosed techniques. Based on these assumptions, the residentialcustomers will benefit from the collaboration programs by cutting theirenergy bills and the DSO will enhance the restoration performance byscheduling the flexibility of behind-the-meter DERs.

FIG. 2 is a flow diagram illustrating an example collaboration frameworkfor power system restoration incorporating diverse distributed energyresources, in accordance with one or more aspects of the presentdisclosure. Specifically, FIG. 2 illustrates one example of thecoordination between the DSO and a HEMS during the restoration stage.The same procedures may apply to all HEMSs subscribing to the DSO'sprograms. The communication involved in the techniques provided hereinmay be achieved using existing communication protocols such as DNP3 andModBus, or other suitable communication means.

As shown in the example of FIG. 2, the DSO manages the systemreconfiguration and utility DER operation, and the HEMS schedules thebehind-the-meter DERs. The flexibility of behind-the-meter DERs is firstmodeled by the HEMS and then shared, via transmission 202, with the DSOfor restoration strategy optimization. To properly monitor theuncertainty factors during the restoration process, the techniquesdescribed herein consider the integration and collaboration of adaptiverestoration planning and real-time operation, the latter of which isrepresented by dotted box 206.

The restoration planning emphasizes reconfiguring the distributionsystem and scheduling its resources—i.e., DERs—to maintain a reliablepower supply for a prolonged period. A rolling horizon MPC technique isused to perform a rolling optimization that enables the DSO to adjustits planning solution with respect to the most accurate forecastavailable.

FIG. 3 is a conceptual diagram illustrating example timing andsequential control actions for power restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure. In the example of FIG. 3, durations 302A and302B (collectively “durations 302”) each represent a prediction horizonused in optimization. Only two of durations 302 are shown in the exampleof FIG. 3, but durations 302 may include additional durations as needed.Each of durations 302 may also be referred to herein as an MPC horizon.Each MPC horizon is subdivided into a number of control horizons,denoted by durations 304A and 304B (collectively “durations 304”).Again, only two of durations 304 are shown in the example of FIG. 3, butdurations 304 may include additional durations. Durations 304 may alsobe referred to herein as MPC intervals. When solving an optimizationmodel on an MPC horizon, the solutions at all MPC intervals are derived,but only the control action at the first MPC interval is executed. Inthe distribution system restoration problem, the MPC interval maycorrespond to a typical control horizon of utility control systems(e.g., 15 minutes or 1 hour). The MPC horizon is influenced by thecapability to forecast renewable DER generation. To ensure high forecastaccuracy, it is suggested to use a short-term forecast that lasts for afew hours.

As indicated by dotted box 206 of FIG. 2, the MPC-based planningsolution acts as a reference for both the DSO and the HEMS duringreal-time operation. Meanwhile, the DER forecast within one MPC intervalcould also fluctuate and threaten the security of the restored network.Therefore, each MPC interval is further divided into several real-timeintervals, denoted as durations 306A and 306B (collectively “durations306”) as shown in FIG. 3. Again, only two of durations 306 are shown inthe example of FIG. 3, but durations 306 may include additionaldurations. Real-time dispatch will be conducted at each real-timeinterval to maintain power balance and guarantee the feasibility of therestoration techniques described herein. Note that throughout thereal-time operation stage (dotted box 206 of FIG. 1), communicationbetween the HEMS and DSO is optional because the HEMS follows only thereceived reference based on the MPC-based planning model, which can nolonger be modified in real-time. Generally, a HEMS should respect thefollowing two principles when scheduling the consumption plans ofvarious behind-the-meter DERs: (1) accommodate the comfort preferencesof individual customers; and (2) protect the customers'private/confidential information from being revealed.

For purposes of exposition, is assumed that each customer might have anuncontrollable load (e.g., lighting load), rooftop photovoltaic (PV)panel, ESS, heat pump, and electric water heater. Different models havebeen developed to capture the unique characteristics of these DERs.Although the DER models are critical for HEMS optimization, a detailedDER modeling technique is not the focus of this disclosure. Therefore, ageneralized model is used instead to concisely summarize existing DERmodels discussed in the literature. The generalized model will beemployed to schedule the behind-the-meter DERs to meet the customers'comfort settings.

For a given HEMS, h, its behind-the-meter DER scheduling model can bedescribed as:

$\begin{matrix}{{Min}\mspace{14mu}{\Sigma_{\tau = T_{st}}^{T_{en}}\left( {{\Sigma_{a \in \Omega_{A_{h}}}p_{h,\tau}^{a}} + {\Sigma_{b \in \Omega_{B_{h}}}\beta_{h}^{b}{{s_{h,\tau}^{b} - s_{h}^{b,{pre}}}}_{2}^{2}}} \right)}} & (1)\end{matrix}$s.t. p _(h,τ) ^(a,min) ≤p _(h,τ) ^(a) ≤p _(h,τ) ^(a,max)  (2)

q _(h,τ) ^(a,min) ≤q _(h,τ) ^(a) ≤q _(h,τ) ^(a,max)  (3)

s _(h,τ) ^(b,min) ≤s _(h,τ) ^(b) ≤s _(h,τ) ^(b,max)  (4)

s _(h,τ) ^(b) =s _(h,τ−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,τ) ^(a) ΔT+c_(h,τ) ^(b) +ΔT  (5)

In this model, the objective function (1) minimizes the total energyconsumption (the first term) and the deviation from preferred comfortsettings (the second term) over the MPC planning horizon from T_(st) toT_(en). A weighting coefficient β_(h) ^(b) is introduced to distinguishthe priority of comfort parameters based on the preferences of thecorresponding customer. A positive (or negative) p_(h,τ) ^(a) is definedherein to indicate that the DER is consuming (or generating) power.

Constraints (2)-(5) are generalized constraints for behind-the-meterDERs. The active power and reactive power of DERs are constrained by (2)and (3), respectively. The customer's comfort parameters are constrainedby (4). Equation (5) captures the influences of DER behaviors on theircorresponding comfort indicator. In (5), δ_(h) ^(a) denotes theefficiency of DER on the comfort of the customer. If a certain comfortparameter is influenced by several DERs, the contributions of DERs willbe aggregated as shown in (5), where a→b indicates that the consumptionof DER a will influence the comfort parameter b. In addition, c_(h,τ)^(b) is introduced to aggregate the influences on the comfort parameterother than DERs.

The DER schedule optimized by model (1)-(5) is denoted as {p_(h,τ)^(a)*}. When interacting with the DSO, each HEMS can reveal itspreferred power consumption schedule {p_(h,τ)*}, where p_(h,τ)* isdefined as

p_(h, τ)^(*): = Σ_(a ∈ Ω_(A_(h)))p_(h, τ)^(a*),

to protect the confidential information of its behind-the-meter DERsfrom the DSO and other HEMSs. In this way, the DSO and other HEMSs donot have access to the DER information such as capacity, key parameters,and consumption pattern. Note that p_(h,τ) ^(a) and q_(h,τ) ^(a) arecontinuous variables in constraints (2) and (3), which is derived fromthe simplified duty-cycle-based DER models. In practice,behind-the-meter DERs could have discrete consumption characteristicsassociated with the on/off status. This issue will be handled in thereal-time dispatch model.

The preferred DER schedules can be obtained by solving model (1)-(5).Moreover, the preferred DER schedules may be modified toincrease/decrease their consumption at selected time intervals toaccommodate system restoration needs while still respecting the comfortconstraint (4). The feasible DER scheduling range is denoted as the DERflexibility. An intuitive approach commonly used to estimate DERflexibility is to calculate the maximum and minimum DER consumptionschedules based on constraints (2)-(5). The DER flexibility obtained bythis intuitive approach cannot accurately reflect the time correlationsof DER schedules. Therefore, the techniques of the present disclosureestimate the DER flexibility by using an energy-based method to accountfor the time correlation illustrated in constraint (5).

For a given HEMS, h, its maximum and minimum energy consumption fromT_(st) to τ (∀τ∈[T_(st), T_(en)]), denoted as e_(h,τ) ^(max) and e_(h,τ)^(min), can be obtained by solving (6) and (7), respectively:

$\begin{matrix}{{{Max}\mspace{14mu} e_{h,\tau}^{\max}} = \left. {\sum\limits_{T = T_{st}}^{\tau}{\sum\limits_{a \in \Omega_{A_{h}}}p_{h,T}^{a}}} \middle| {{s.t.(2)} - (5)} \right.} & (6) \\{{{Min}\mspace{14mu} e_{h,\tau}^{\min}} = \left. {\sum\limits_{T = T_{st}}^{\tau}{\sum\limits_{a \in \Omega_{A_{h}}}p_{h,T}^{a}}} \middle| {{s.t.(2)} - (5)} \right.} & (7)\end{matrix}$

Based on the obtained e_(h,τ) ^(max) and e_(h,τ) ^(min) the flexibilityrange of HEMS h throughout the MPC horizon [T_(st), T_(en)] can bedescribed by the following constraint:

$\begin{matrix}{e_{h,\tau}^{\min} \leq {\Sigma_{T = T_{st}}^{\tau}\Sigma_{a \in \Omega_{A_{h}}}p_{h,T}^{a}} \leq e_{h,\tau}^{\max}} & (8)\end{matrix}$

Note that the DER capacity limit (2) is still valid to guarantee thefeasibility of the developed DER flexibility range. To accommodate thecustomers' privacy protection requirement, the original constraint (2)will be reformulated as:

$\begin{matrix}{p_{h,\tau}^{\min} \leq {\Sigma_{a \in \Omega_{A_{h}}}p_{h,\tau}^{a}} \leq p_{h,\tau}^{\max}} & (9)\end{matrix}$

where e_(h,τ) ^(max) and e_(h,τ) ^(min) will be calculated by adding thelimits of individual DERs (i.e., e_(h,τ) ^(a,max) and e_(h,τ) ^(a,max)).

The feasibility of this energy-based method is proved below.

Theorem: The feasible region based on the intuitive approach is a propersubset of the feasible region based on the energy-based method describedbelow.

Proof: Denote the feasible regions based on the intuitive approach andthe proposed energy-based method as

^(I) and

^(II), respectively. If

^(I) is a proper subset of

^(II), i.e.,

^(I)

^(II),

^(I) must be a subset of

^(II) and does not equal

^(II). For any feasible solution p_(h,τ) ^(a)∈

^(I), constraint (2) is accommodated VT. Thus, the following constraintholds:

$\begin{matrix}{{\Sigma_{t = T_{st}}^{\tau}\Sigma_{a \in \Omega_{A_{h}}}p_{h,t}^{a,\min}} \leq {\Sigma_{t = T_{st}}^{\tau}\Sigma_{a \in \Omega_{A_{h}}}p_{h,t}^{a}} \leq {\Sigma_{t = T_{st}}^{\tau}\Sigma_{a \in \Omega_{A_{h}}}p_{h,t}^{a,\max}}} & ({P1})\end{matrix}$

Note that (P1) is equivalent to (8). Hence, p_(h,τ) ^(a) satisfies both(2) and (8). Therefore, p_(h,τ) ^(a)∈

^(II). Thus,

^(I) ⊆

^(II) is proved.

On the contrary, select two adjacent time slots T₁, τ₂ and let τ₂=τ₁+1.Construct a solution p_(h,τ) ^(a) based on the following (P2)-(P4).Apparently, ∃Δx>0 that guarantees p_(h,τ) ^(a) accommodates (8) as well.Therefore, p_(h,τ) ^(a) is feasible according to the proposedenergy-based method, i.e., p_(h,τ) ^(a)∈

^(II).

p _(h,τ) ^(a) =p _(h,τ) ^(a,min),∀τ,τ≠τ₁,τ≠τ₂  (P2)

p _(h,τ) ₁ ^(a) =p _(h,τ) ₁ ^(a,min) +Δx  (P3)

p _(h,τ) ₂ ^(a) =p _(h,τ) ₂ ^(a,min)−0.5Δx  (P4)

However, P_(i,d,τ) ^(D(II)) is not a feasible solution according to theintuitive approach because constraint (2) is violated at time π₂, i.e.,p_(h,τ) ^(a)∉

^(I). As a result,

^(I) does not equal

^(II) and

^(I)⊆

^(II) is proved.

In summary, each HEMS will use {e_(h,τ) ^(min),e_(h,τ) ^(max),p_(h,τ)^(min),p_(h,τ) ^(max)} to represent its flexibility. Combined with thepreferred schedule {p}, the confidential information of thecustomers-such as parameters of behind-the-meter DERs and comfortpreferences-will not be exposed to the DSO or other HEMSs.

After an outage, distribution system restoration may be achieved throughreconfiguring the system into several islands. In general, thedistribution system reconfiguration problem considers only balancedsystems and may be solved using mathematical programming methods andinteger variables. In the techniques of the present disclosure, whereMPC is employed and the three-phase unbalanced system is considered, thecomputational performance of conventional approaches may beinsufficient. Therefore, in some examples, the three-phase distributionsystem restoration model may be linearized using the method developed inL. Gan and S. H. Low, “Convex relaxations and linear approximation foroptimal power flow in multiphase radial networks,” published in PowerSystems Computation Conference, Wroclaw, Poland, August 2014(hereinafter “Gan et al.”), the relevant portions of which areincorporated herein by reference for such purposes. In such examples,the distribution system reconfiguration solution may be solved through aheuristic process which will be described below. The three-phaselinearized system restoration model is described as:

Min Σ_(τ=T) _(st) ^(T) ^(en) Σ_(n∈Ω) _(N) [γ_(n) ^(G) P _(n,τ)^(G)+γ_(n) ^(S)Σ_(ϕ∈Ψ) _(n) (P _(n,τ) ^(S,ϕ) +Q _(n,τ) ^(S,ϕ))]+Σ_(τ=T)_(st) ^(T) ^(en) Σ_(g∈Ω) _(H) (γ_(h) ^(V) P _(h,τ) ^(V)+γ_(h) ^(W) P_(h,τ) ^(W))  (10)

s.t. P _(n,τ) ^(G,min) ≤P _(n,τ) ^(G) ≤P _(nτ) ^(G,max)  (11)

Q _(n,τ) ^(G,min) ≤Q _(n,τ) ^(G) ≤Q _(nτ) ^(G,max)  (12)

P _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) P _(n,τ) ^(G,ϕ) ,Q _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) Q_(n,τ) ^(G,ϕ)  (13)

P _(n,τ) ^(G,ϕ) −P _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (P_(nm,τ) ^(ϕ)−_(mn,τ) ^(ϕ))  (14)

Q _(n,τ) ^(G,ϕ) −Q _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (Q_(nm,τ) ^(ϕ) −Q _(mn,τ) ^(ϕ))  (15)

$\begin{matrix}{P_{n,\tau}^{D,\phi} = {P_{n,\tau}^{D,{fix},\phi} + {\Sigma_{h\rightarrow{\langle{n,\phi}\rangle}}P_{h,\tau}}}} & (16) \\{Q_{n,\tau}^{D,\phi} = {Q_{n,\tau}^{D,{fix},\phi} + {\Sigma_{h\rightarrow{\langle{n,\phi}\rangle}}Q_{h,\tau}}}} & (17)\end{matrix}$

0≤P _(n,τ) ^(S,ϕ) ≤P _(n,τ) ^(D,ϕ),0≤Q _(n,τ) ^(S,ϕ) ≤Q _(n,τ)^(D,ϕ)  (18)

V _(n,τ) ^(ϕ) −V _(m,τ) ^(ϕ)=2(r _(nm) ^(ϕϕ) P _(nm,τ) ^(ϕ) +x _(nm)^(ϕϕ) Q _(nm,τ) ^(ϕ))−2ϑ₁(r _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″)P _(nm,τ) ^(ϕ″) +x _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +x _(nm) ^(ϕϕ″) Q_(nm,τ) ^(ϕ″))−2ϑ₂(x _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) −x _(nm) ^(ϕϕ″) P_(nm,τ) ^(ϕ″) −r _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″) Q _(nm,τ)^(ϕ″))  (19)

V _(n) ^(ϕ,min) ≤V _(n,τ) ^(ϕ) ≤V _(n) ^(ϕ,max)  (20)

P _(h,τ) ^(W) ≥P _(h,τ) −p _(h,τ)*  (21)

P _(h,τ) ^(W) ≥p _(h,τ) *−P _(h,τ)  (22)

p _(h,τ) ^(min) ≤P _(h,τ) ≤p _(h,τ) ^(max)  (23)

e _(h,τ) ^(min) −P _(h,τ) ^(V)≤Σ_(T=T) _(st) ^(τ) P _(h,T) ≤e _(h,τ)^(max) P _(h,τ) ^(V)  (24)

P _(h,τ) ^(V)≥0  (25)

The objective function (10) minimizes the weighted sum of the generationcost, the load-shedding penalty, the HEMS desired schedule deviationpenalty, and the HEMS flexibility range violation penalty. In (10),γ_(n) ^(G) indicates the DER generation cost and γ_(n) ^(S) denotes loadshedding penalty. To reduce renewable energy curtailment duringrestoration, negative generation cost γ_(n) ^(G) may be assigned torenewable DERs and positive γ_(n) ^(G) may be assigned to conventionalgenerators such as diesel generators. Besides, load restoration istypically the most important target during distribution systemrestoration, thus γ_(n) ^(S)>>|γ_(n) ^(G)|. In this way, renewableenergy will be prioritiezed for restoring load supply. If renewableenergy generation is insufficient, conventional generators will beimplemented to reduce load curtailment. γ_(n) ^(V) and γ_(n) ^(W) denotethe penalty coefficients for HEMS comfort deviations and are decided bythe DSO-HEMS collaboration program described above.

The utility-scale DER generation outputs are constrained in (11) and(12). Equation (13) calculates the power output of a utility-scale DERby totaling its generation at all phases. The three-phase nodalinjection equations are represented by (14) and (15), where the powerlosses are ignored for simplicity. The nodal power loads are calculatedby (16) and (17), where h→

n, ϕ

indicates that household h is connected to phase ϕ at bus n. Constraint(18) ensures that load shedding cannot exceed the total demand. Thethree-phase voltage drop is calculated by a linearized equation (19)based on Gan et al. In (19), ϕ′ and ϕ″ denote the phase that leads andlags ϕ by 2π/3, respectively. Moreover, ε₁ and ϑ₂ denote cos(2π/3) andsin(2π/3), respectively. The voltage magnitude is constrained by (20).

Constraints (21)-(25) are introduced to model the flexibility ofbehind-the-meter DERs. The optimized HEMS schedule may deviate from itspreferred schedule and the deviation P_(h,τ) ^(W) is calculated by (21)and (22). Note that P_(h,τ) ^(W) is non-negative, which is guaranteed by(21) and (22). Constraint (23) ensures that the power schedule fallswithin the capacity limit of the HEMS concerned. The flexibility ofbehind-the-meter DERs is constrained by (24). If the cumulative energyconsumption from T_(st) to τ is not within [e_(h,τ) ^(min), e_(h,τ)^(max)], the comfort preference cannot be satisfied, and a positiveP_(h,τ) ^(V) will be obtained by (24) to penalize the DSO. Note thatP_(h,τ) ^(V) is also non-negative, which is enforced by (25).

In summary, the distribution system restoration model (10)-(25) is alinear programming model that employs the flexibility of DERs to satisfyuncontrollable load supply throughout the MPC horizon from T_(st) toT_(en). Note that the DSR model optimizes the schedules of bothutility-scale DERs and individual HEMS. Ideally, each utility-scale DERand HEMS will follow its optimal power schedule, respectively denoted asP_(n,τr) ^(G)* and P_(h,τ)*, in the real-time dispatch. However, becauseof inevitable forecast errors introduced by the intermittency ofrenewable energy sources and arbitrary demand behaviors, the optimal MPCsolution may be invalid in real-time operation, and thus extra measuresmay be necessary to maintain reliable power supply after an outage.

The three-phase distribution system power flow model does not considertopological changes through switch operations. Typically, mathematicalprogramming approaches will be employed to reconfigure distributionsystems with the help of binary variables denoting the status ofswitches. For instance, in the present example, a mathematicalprogramming approach may involve the introduction of binarydecision-making variables as follows:

κ_(n) Binary variable of bus restoration status

ξ_(em) Binary variable of line restoration status

S_(nm) ^(ϕ,max) Capacity of distribution line

M A sufficiently large positive scalar

Accordingly, the nodal power balance (11)-(12) and voltage constraint(20), above, would be modified as follows:

κ_(n) P _(n,τ) ^(G,min) ≤P _(n,τ) ^(G)≤κ_(n) P _(n,τ) ^(G,max)  (11*)

κ_(n) P _(n,τ) ^(G,min) ≤Q _(n,τ) ^(G)≤κ_(n) Q _(n,τ) ^(G,max)  (12*)

κ_(n) V _(n) ^(ϕ,min) ≤V _(n,τ) ^(ϕ)≤κ_(n) V _(n) ^(ϕ,max)  (20*)

Additional constraints would need to be added to constrain the lineflow, as well:

(P _(nm,τ) ^(ϕ))²+(Q _(nm,r) ^(ϕ))²≤ξ_(nm)(S _(nm) ^(ϕ,max))²

κ_(n) M≥Σ _(∀nm∈Ω) _(L) ξ_(nm) ,∀n∈Ω _(N)

κ_(n)∈{0,1},∀n∈Ω _(N)

ξ_(nm)∈{0,1},∀nm∈Ω _(L)

Note that κ_(n) and ξ_(nm) are not time-variant according to the aboveconstraints, which indicates that the restoration islands remainunchanged throughout the considered horizon. This is because a stabletopology is needed to ensure a reliable and sustainable power supply,which is desirable after a power outage.

In the mathematical programming approach, topology constraints should beenforced to guarantee the restored distribution systems are operating inradial fashion.

ξ_(nm) ^(F) Lines that have suffered from faults/failures

μ_(nm) ⁺, μ_(nm) ⁻ Direction of virtual flow in distribution lines

λ_(n) Binary variable indicating virtual flow injection

μ_(n,τ) ^(VS), p_(n,τ) ^(VD) Virtual power generation and consumption

p_(nm,τ) ^(V) Virtual power flow

A new set of radiality constraints is developed by modifying thetraditional spanning-tree constraints to ensure the radiality of areconfigured distribution system with a variable number of islands. Thetraditional spanning-tree constraints can be described as:

μ_(nm) ⁺+μ_(nm) ⁻=γ_(nm) ,∀nm∈Ω _(L)  (MP0)

μ_(nm) ⁺≥0,μ_(nm) ⁻≥0,∀nm∈Ω _(L)  (MP1)

Σ_(m∈Ω) _(N) μ_(nm) ⁺ =,∀n∈Ω _(CG)∪Ω_(RG)  (MP2)

Σ_(m∈Ω) _(N) μ_(nm) ⁺=1,∀n∈Ω _(N)\{Ω_(CG)∪Ω_(RG)}  (MP3)

Equation (MP0) constrains that virtual flow cannot flow through openeddistribution lines. The virtual flow directions are non-negative asshown in (MP1). Constraint (MP2) enforces that the source buses have novirtual flow injection. For other buses, virtual flow is injected from aunique feeder as constrained in (MP3).

However, the spanning tree constraints (MP0)-(MP3) have the followinglimitations: (1) all the buses in the distribution system should berestored, which does not always work when power generation capacity isnot sufficient to supply all the loads; and (2) each divided island hasa unique source bus, i.e., the number of restoration islands equals thenumber of source buses.

For the first limitation, unrestored load buses should be enabled whenoptimizing the DSR. If a bus is not restored, no virtual flow will beinjected into that bus as shown in (MP4):

Σ_(m∈Ω) _(N) μ_(nm) ⁺=κ_(n) ,∀n∈Ω _(N)\{Ω_(CG)∪Ω_(RG)}  (MP4)

Moreover, the number of islands should be a variable to overcome thesecond limitation. Referring to graph theory, if a distribution systemshould be divided into N^(I) radial subnetworks, the followingrelationship between the number of connected buses and the number ofrestored lines can be established:

Σ_(n∈Ω) _(N) κ_(n)−ΣΣ_(nm∈Ω) _(L) ξ_(nm) =N ^(I)  (MP5)

As for a distribution system with N^(S) source buses at time t, thenumber of formulated islands should not exceed the number of sourcebuses as shown in (MP6).

1≤N ^(I) ≤N ^(S)  (MP6)

Typically, substations and buses with large controllable generationassets are regarded as the source buses to provide power supply andmaintain stability. Nowadays, inverter-based DERs may also have thegrid-forming capability with appropriate control strategies. Therefore,the buses of substations and all utility-managed DERs are assumed to becandidate source buses. If a DER does not have the grid-formingcapability, it will be excluded from Ω_(CG)∪Ω_(RG). Aggregator-managedDERs are not included because their capacities are generally limited.

To accommodate the possibility of integrating multiple source buses intoone island, an ancillary binary variable λ_(n) is introduced to modifythe original spanning tree constraints:

Σ_(m∈Ω) _(N) μ_(nm) ⁺=λ_(n) ,∀n∈Ω _(CG)∪Ω_(RG)  (MP7)

Σ_(n∈Ω) _(N) λ_(N) =N ^(S) −N _(I) ,∀n∈Ω _(CG)∪Ω_(RG)  (MP8)

λ_(n)∈{0,1},∀n∈Ω _(CG)∪Ω_(RG)  (MP9)

Compared to (MP2), virtual flow injection is enabled at source buseswith (MP7). If N^(I) islands are formulated, the number of source buseswith virtual flow injection is calculated by (MP8). Combining(MP7)-(MP9), the modified spanning tree constraints allow a variablenumber of islands, and each island may contain multiple source buses.Note that the virtual flow directions in the modified (MP7)-(MP9) nolonger indicate the directions of actual power flows in the distributionsystem.

An additional constraint is introduced to accommodate possible failuresin distribution lines. Constraint (MP10) enforces that distributionlines that are suffering from faults will not be restored.

ξ_(nm)≤1−ξ_(nm) ^(F) ,∀nm∈Ω _(L)  (MP10)

However, the modified spanning tree constraints alone do not guaranteethe radiality at all islands, especially with multiple DERs acting assource buses. In the present disclosure, virtual flow balance isintroduced to guarantee the connectivity within each divided island asdescribed below:

p _(n,ξ) ^(VS) −p _(n,τ) ^(VD)=Σ_(nm∈Ω) _(L) (p _(nm,ξ) ^(V) −p _(mn,ξ)^(V))  (MP11)

0≤p _(n,τ) ^(VS)≤(1−λ_(n))M  (MP12)

p _(n,τ) ^(VD)≥κ_(n)  (MP13)

0≤p _(nm,τ) ^(V)≤ξ_(nm) M  (MP14)

Equation (MP11) denotes the nodal virtual power injection balance.Constraint (MP12) guarantees that only the source buses with no virtualflow injections can supply virtual power. Constraint (MP13) ensures thatall restored buses have a virtual power load. Constraint (MP14) enforcesthat virtual power can only flow in restored lines. The big-M method isalso employed in (MP12) and (MP14), where M equals the total number ofbuses in the system.

With the complementary constraints (MP 11)-(MP14), the connectivity andradiality of every island can be ensured. In summary, the modifiedradiality constraints contain (MP0)-(MP1) and (MP4)-(MP14).

Compared with existing radiality constraints, the provided constraintscan accommodate a variable number of radial islands to account forsystem conditions and DER capabilities while introducing a minimumnumber of binary variables. The required binary variables based on theseconstraints are κ_(n) (∀n∈Ω_(N)), ξ_(nm) (∀nm∈Ω_(L)), and λ_(n)(∇∀n∈Ω_(CG)∈Ω_(RG)). Hence, the number of required binary variables ofthe mathematical programming method is |N|+|L|+|S|, where |N|, |L|, and|S| denote the number of buses, number of lines, and the number ofsources buses, respectively.

The introduction of binary variables can significantly increase thecomputational burden, however. In accordance with the providedtechniques (i.e., FIG. 2), where the restoration planning model will besolved repeatedly (iteratively) using the MPC technique, thecomputational efficiency plays an important role in guaranteeing thefeasibility of the proposed strategy. Therefore, the techniquesdescribed herein utilize a fast heuristic algorithm provided in Q. Peng,Y. Tang, and S. H. Low, “Feeder reconfiguration in distribution networksbased on convex relaxation of OPF,” published in IEEE Trans. PowerSyst., vol. 30, no. 4, pp. 1793-1804, July 2015 (hereinafter “Peng etal.”), the relevant portions of which are incorporated herein byreference for such purposes. This fast heuristic algorithm is employedand integrated into the adaptive restoration planning model toefficiently obtain switch configurations.

FIG. 4 is a flow diagram illustrating example operations for networkreconfiguration during power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure. In addition to reducing computational burden,the employed heuristic algorithm can effectively handle extreme outagescenarios such as ill-conditioned network and loss of parts of thenetwork as the restoration planning model (10)-(25) will be executed ateach isolated subnetwork. A detailed description of the fast heuristicalgorithm can be found in Peng et al. and will not be further discussedhere.

As shown in the example of FIG. 3, each MPC interval (i.e., each ofdurations 304) is further divided into n_(t) real-time intervals (i.e.,durations 306) to handle the forecast error and expected fluctuationswithin a single MPC interval. The optimized MPC schedules—i.e. p_(n,τ)^(G)* and P_(h,τ)*—will be employed as reference points for DSOs andHEMS in real-time operation, respectively. Thus, as shown in FIG. 2, theDSO will provide the optimized schedule, via transmission 204, toparticipating HEMSs.

In real-time, the DERs will be solely controlled by their correspondingHEMS for privacy protection. Because P_(h,τ)* is the power consumptionreference of an entire household, each HEMS must control the schedulesof its behind-the-meter DERs to match the total consumption reference inreal-time. If the HEMS fails to match the received reference schedule,it might be penalized according to the program to which it subscribedfor putting extra burden on the DSO to maintain real-time power balance.

As discussed previously, the rewards of participating demand responseprograms are fixed payments and do not affect optimization models. Interms of penalties, the following two are introduced as part of thetechniques described herein: (i) If the difference between P_(h,τ)* andthe total HEMS consumption,

${\sum\limits_{a \in \Omega_{A_{h}}}p_{h,\tau}^{a}},$

at any real-time interval ω, is larger than an acceptable threshold, η,the HEMS is subject to penalty. (ii) The HEMS will be penalized for thedeviation of total real-time energy consumption and reference energyconsumption during the MPC interval ΔT—i.e., the HEMS will pay for thedifference between P_(h,τ)*ΔT and

Σ_(ω)Σ_(a ∈ Ω_(A_(h)))p_(h, ω)^(a)Δt.

Based on these two penalties, the real-time control model of DERs can bedescribed as:

s.t. u _(h,ω) ^(a) p _(h,ω) ^(a,min) ≤p _(h,ω) ^(a) ≤u _(h,ω) ^(a) p_(h,ω) ^(a,max)  (27)

u _(h,ω) ^(a) q _(h,ω) ^(a,min) ≤q _(h,ω) ^(a) ≤u _(h,ω) ^(a) q _(h,ω)^(a,max)  (28)

s _(h,ω) ^(b,min) ≤s _(h,ω) ^(b) ≤s _(h,ω) ^(b,max)  (29)

s _(h,ω) ^(b) =s _(h,ω−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,ω) ^(a) Δt+c_(h,ω) ^(b) Δt  (30)

$\begin{matrix}{{c_{h,\omega}^{P} \geq {{\sum_{a \in \Omega_{A_{h}}}p_{h,\omega}^{a}} - {\left( {1 + \eta} \right)P_{h,\tau}^{*}}}},{{{if}\mspace{14mu} P_{h,\tau}^{*}} \geq 0}} & (31) \\{{c_{h,\omega}^{P} \geq {{\left( {1 - \eta} \right)P_{h,\tau}^{*}} - {\sum_{a \in \Omega_{A_{h}}}p_{h,\omega}^{a}}}},{{{if}\mspace{14mu} P_{h,\tau}^{*}} \geq 0}} & (32) \\{{c_{h,\omega}^{P} \geq {{\sum_{a \in \Omega_{A_{h}}}p_{h,\omega}^{a}} - {\left( {1 - \eta} \right)P_{h,\tau}^{*}}}},{{{if}\mspace{14mu} P_{h,\tau}^{*}} < 0}} & (33) \\{{c_{h,\omega}^{P} \geq {{\left( {1 + \eta} \right)P_{h,\tau}^{*}} - {\sum_{a \in \Omega_{A_{h}}}p_{h,\omega}^{a}}}},{{{if}\mspace{14mu} P_{h,\tau}^{*}} < 0}} & (34) \\{c_{h,\omega}^{e} \geq {\left( {n_{t} - \omega + 1} \right)^{- 1}{\sum_{\sigma = 1}^{\omega}\left( {{\sum_{a \in \Omega_{A_{h}}}p_{h,\sigma}^{a}} - P_{h,\tau}^{*}} \right)}}} & (35) \\{c_{h,\omega}^{e} \geq {\left( {n_{t} - \omega + 1} \right)^{- 1}{\sum_{\sigma = 1}^{\omega}\left( {P_{h,\tau}^{*} - {\sum_{a \in \Omega_{A_{h}}}p_{h,\sigma}^{a}}} \right)}}} & (36)\end{matrix}$Σ_(σ=ω−1−ω) _(h) _(a,off) ^(ω−1)(1−u _(h,σ) ^(a))≥ω_(h) ^(a,off)(u_(h,ω) ^(a) −u _(h,ω−1) ^(a))  (37)

Σ_(σ=ω−1−ω) _(h) _(a,off) ^(ω−1) u _(h,σ) ^(a)≥ω_(h) ^(a,on)(u _(h,ω−1)^(a) −u _(h,ω) ^(a))  (38)

u _(h,ω) ^(a)∈{0,1}  (39)

The objective function (26) minimizes the weighted sum of two real-timepenalties and the deviation from preferred comfort settings. In (26),β_(h) ^(P), β_(h) ^(e), and β_(h) ^(b) are the weighting factors for thedeviation of power, deviation of consumed/generated energy for thecorresponding period, and the deviation of preferred comfort settings,respectively. β_(h) ^(P) and β_(h) ^(e) are decided by the DSO-HEMScollaboration program and indicate the actual financial penalties forthe HEMS if it fails to meet the received setpoints. β_(h) ^(b)indicates the preferences of the HEMS and may vary from one HEMS toanother. If a customer prioritizes the economic profit (or comfort),β_(h) ^(b) will be assigned a small (or large) value, e.g., β_(h)^(b)→0(β_(h) ^(b)→∞).

Constraints (27)-(30) are slightly different from (2)-(5) due to theintroduction of the Boolean DER control signal, u_(h,ω) ^(a). Thedifference between the real-time power schedule and the reference poweris calculated by (31)-(34). Note that either (31)-(32) or (33)-(34) willbe implemented depending on the value of P, Because P, is optimized bythe DSO and is known to the HEMS when solving the real-time controlmodel, constraints (31)-(34) are still linear constraints. However, thetotal energy consumption deviation cannot be obtained before solving alln_(t) time intervals. Hence, constraints (35)-(36) are employed toestimate the cost of the energy mismatch. In (35)-(36), the coefficient(n_(t)−ω+1)⁻¹ is introduced to assign different weights based on thecurrent real-time interval. At the beginning of the MPC interval,(n_(t)−ω+1)⁻¹ is small, indicating that the energy mismatch for thefirst ω real-time intervals can be easily compensated in the remainingreal-time intervals. On the other hand, (n_(t)−ω+1)⁻¹ is increasing whenapproaching the end of the MPC interval, meaning that mitigating thesame amount of energy mismatch will require more effort because of thelimited remaining time. Hence, constraints (35)-(36) can guarantee aflexible control schedule for DERs at the beginning real-time intervalsand ensure a minimal energy consumption deviation for the entire MPCinterval.

The Boolean control signal, u_(h,ω) ^(a), is constrained by (37)-(39).In practice, customers may not prefer to frequently alter the on/offstatus of DERs. To capture this characteristic, it can be assumed thateach household appliance has a minimum on/off time and the controlsignal u_(h,ω) ^(a) can follow the similar principles used to constrainthe on/off of thermal power plants in unit commitment models. In thisway, the appliances will not be turned on (or off) before the minimumoff- (or on-) time is met. When solving the real-time model at time ω,the only variable in constraints (37) and (38) is u_(h,ω) ^(a). PreviousDER control signals, such as u_(h,ω−1) ^(a), are known parameters.Compared with the simplified duty cycle model mentioned above, whichassumes that DER consumption can be continuously controlled, morerealistic DER controls can be obtained by solving the proposed real-timecontrol model with u_(h,ω) ^(a) and (37)-(39).

The optimal solution of the real-time model (26)-(39), denoted asp_(h,ω) ^(a⊕), is the final DER setpoint that will be implemented. Atreal-time interval ω, the difference between P_(h,ω)*, and the totalHEMS power consumption p_(h,ω) ^(⊕), which is defined as

p_(h, ω)^(⊕) := Σ_(a ∈ Ω_(A_(h)))p_(h, ω)^(a, ⊕),

will be managed by the DSO. It is assumed that the DSO can acquirep_(h,ω) ^(⊕) from real-time measurements and communication systems asdiscussed above, and it will not be further elaborated on herein, forbrevity.

During real-time operation, the DSO must deal with volatile renewableenergy generation, forecast error of uncontrollable demand, and thereal-time deviations of HEMS to maintain power balance. It is assumedthat the following resources may be available to the DSO in real-timeoperation: (i) Each conventional DER has a limited regulation capacitythat can provide fast ramping up/down in real-time. (ii) Each renewableDER has a small ESS to level off its variable generation output. Thesmall ESS does not participate in the MPC-based DSR model describedabove.

The DSO may use these resources to mitigate power unbalance in real-timeoperation. The real-time distribution system dispatch model is describedas follows:

Min Σ_(n∈Ω) _(N) (γ_(n) ^(S) P _(n,ω) ^(S)+γ_(n) ^(C) P _(n,ω)^(curt)+γ_(n) ^(ES)∥ρ_(n,ω) ^(ES)−ρ_(n,ω) ^(ES,pre)∥₂ ²)  (40)

s.t. P _(n,τ) ^(G) *−P _(n,τ) ^(G,reg) ≤P _(n,ω) ^(G) ≤P _(n,τ) ^(G) *+P_(n,τ) ^(G,reg) ,∀n∈Ω _(CG)  (41)

P _(n,ω) ^(G) =P _(n,ω) ^(G,act) +P _(n,ω) ^(ES,D) −P _(n,ω) ^(ES,C) −P_(n,ω) ^(curt) ,∀n∈Ω _(RG)  (42)

ρ_(n,ω) ^(ES)=ρ_(n,ω−1) ^(ES)+(δ_(n) ^(C) P _(n,ω) ^(ES,C)−δ_(n) ^(D) P_(n,ω) ^(ES,D))Δt,∀n∈Ω _(RG)  (43)

ρ_(n,ω) ^(ES,min)≤ρ_(n,ω) ^(ES)≤ρ_(n,ω) ^(ES,max) ,∀n∈Ω _(RG)  (44)

0≤P _(n,ω) ^(ES,C) ≤P _(n) ^(ES,C,max),0≤P _(n,ω) ^(ES,D) ≤P _(n)^(ED,D,max) ,∀n∈Ω _(RG)  (45)

$\begin{matrix}{P_{n,\omega}^{D,\phi} = {P_{n,\omega}^{D,{fix},\phi} + {\Sigma_{h\rightarrow{\langle{n,\phi}\rangle}}p_{h,\omega}^{\oplus}}}} & (46)\end{matrix}$

$\begin{matrix}{Q_{n,\omega}^{D,\phi} = {Q_{n,\omega}^{D,{fix},\phi^{\prime}} + {\sum_{h\rightarrow{\langle{n,\phi}\rangle}}q_{h,\omega}^{\oplus}}}} & (47) \\{{0 \leq P_{n,\omega}^{S,\phi} \leq P_{n,\omega}^{D,\phi}},{0 \leq Q_{n,\omega}^{S,\phi} \leq Q_{n,\omega}^{D,\phi}}} & (48)\end{matrix}$

-   -   and constraints (12)-(15), (19), and (20)

The objective function (40) minimizes the weighted real-timeload-shedding penalty, renewable energy curtailment, and the usage ofsmall ESS that are associated with distributed renewable generators. Thegeneration output of a conventional distributed generator is constrainedby (41), indicating that the real-time generation cannot deviate fromits reference, P_(n,ω) ^(G)* larger than its regulation capacity,P_(n,ω) ^(G,reg). The generation of a distributed renewable generator isconstrained by (42). The real-time dispatch of the ESS is constrained by(43)-(44). The ESS charging and discharging power are constrained by(45). Note that ρ_(n,ω−1) ^(ES) is a known parameter when solving thereal-time model at time ω. Constraints (46)-(48) are similar to(16)-(18); however, the HEMS real-time consumption—i.e., p_(h,ω) ^(⊕)and q_(h,ω) ^(⊕) are no longer variables in (46)-(48). The distributionsystem operational constraints (12)-(15), (19), and (20) should also beintegrated into the real-time dispatch model. Note that (11) has alreadybeen replaced by (41) and (42) in the real-time model.

The MPC-based restoration planning models and real-time dispatch modelsused by the techniques described herein were developed above.Considering the temporal correlation illustrated in the example of FIG.3, an example algorithm is designed to integrate restoration planningand real-time control. The detailed operations are listed in Algorithm1.

Algorithm 1

-   -   0: The algorithm is initialized if the DSO detects an outage.        Service collaboration requests will be sent to HEMSs that        subscribed to the restoration program.    -   1: The DSO and HEMSs will each initialize the MPC interval τ←1    -   2: Each HEMS solves its optimization model (1)-(5) over the MPC        time horizon [T_(st)+τ−1, T_(en)+τ−1], estimates the DER        flexibility range based on (6)-(9), and shares the DER        flexibility data and its preferred energy schedule with the DSO.    -   3: DSO optimizes distribution system restoration planning model        (10)-(25) over the MPC time horizon [T_(st)+τ−1, T_(en)+τ−1]        based on fault data, DER generation forecasts, and DER        flexibility data shared by HEMSs. The reconfiguration algorithm        described in FIG. 4 will be implemented to ensure radiality.    -   4: DSO sends the optimal solution of MPC interval T to        utility-owned DERs and HEMSs as the operating setpoints.    -   5: Initialize real-time interval ω←1 for the current MPC        interval τ    -   6: Each HEMS optimizes its real-time dispatch at real-time        interval ω based on model (26)-(39) and received setpoints for        MPC interval τ.    -   7: DSO monitors system status at real-time interval ω and        deploys necessary control measures based on the model developed        in Section IV.B.    -   8: If ω<n_(t), the DSO and HEMSs will each return to Operation 6        with ω←ω+1.    -   Otherwise, the DSO and HEMSs will each return to Operation 2        with τ←τ+1

As shown in the example of Algorithm 1, operations 1-4 will generate theMPC-based distribution system restoration planning solution andoperations 5-7 are employed to finalize the real-time dispatch for bothbehind-the-meter and utility-scale DERs.

In terms of computational burden, operations 2 and 6 can be performed inparallel because the HEMS models are decoupled. Thus, the computationaltime for MPC-based restoration planning is the longest HEMS optimizationtime in operation 2 plus the total time spent in operation 3 forreconfiguration. Similarly, the computational time for the real-timeoperation is the longest HEMS optimization time in operation 6 plus thetime spent in operation 7. Note that operation 7 does not necessarilyneed the inputs from operation 6. In other words, operations 6 and 7 canbe executed in parallel as well. However, it was still assumed thatoperation 7 will not start before operation 6 calculation is finishedwhen evaluating the computational performance of Algorithm 1.

In Algorithm 1, the distribution system reconfiguration is implementedin operation 3, meaning that line switching operations only occur at theMPC time scale. For example, if a line outage or repair occurs duringthe restoration process, the network information will be updated at thenext MPC interval for topology reconfiguration. This coincides withexisting utility practice where frequent line switching is prohibited.Therefore, this disclosure does not consider reconfiguration in thereal-time operation stage, although it is readily feasible to integratethe reconfiguration algorithm depicted in FIG. 4 into the real-timeoperations.

FIG. 5 is a conceptual diagram illustrating an example simplified powernetwork (network 502) that may employ power system restorationincorporating diverse distributed energy resources, in accordance withone or more aspects of the present disclosure. A simple demonstration isgiven below within the context of FIG. 2 to explain the interactionbetween the various aspects of the techniques described herein. As shownin FIG. 2, network 502 includes distribution network 503 (e.g., the restof the distribution network) and buses 504A, 504B, and 504C(collectively “buses 504”). Assume that the didactical system with buses504 is isolated from distribution network 503, as shown in FIG. 5. Theisolated system has utility-scale PV installation 505 (connected to bus504A), HEMS 506 (connected to bus 504B), and uncontrollable load 507(connected to bus 504C). In this example, uncontrollable load 507 has afixed 10 kW consumption. The isolated system also contains a loop.

For simplicity, in this didactical example, assume the MPC interval is1-hour, and the MPC horizon is 2 hours, i.e., two MPC intervals (τ₁ andτ₂) are considered. Each MPC interval is divided into two 30-minreal-time intervals (ω₁ and ω₂). The operation of the techniquesdescribed herein for power system restoration incorporating diversedistributed energy resources can be described as follows:

-   -   HEMS 506 optimizes its energy consumption as described in        Algorithm 1, operation 2. Assume this HEMS respectively needs at        least 0 kWh and 2 kWh by the end of τ₁ and τ₂ to maintain its        comfort (i.e., e₁ ^(min)=0 kWh and e₂ ^(min)=2 kWh). It is also        assumed that the other DER flexibility range constraints such as        maximum energy consumption will not bound so they are ignored in        this example.    -   The DSO (not shown) solves the restoration planning model for        the isolated system. One switch, e.g., a switch between buses        504A and 504C, will be opened to maintain radiality according to        Algorithm 1, operation 3. Assume the PV generation at T₁ and the        forecast for T₂ are respectively 12 kWh and 11 kWh. The optimal        solution is to assign the 2 kWh HEMS demand to the T₁ where PV        generation is abundant.    -   Upon receiving the setpoint for the first MPC interval, HEMS 506        will employ the real-time dispatch model to control its DERs        (Algorithm 1, operation 6). Assume the HEMS has one heat pump        (not shown) with a rating capacity of 4 kW and the heat pump        will be turned on at ω₁. To minimize the energy consumption        deviation (i.e., the second penalty in Section IV.A), the heat        pump will be turned off at ω₂, thus the total consumption for        the first MPC interval remains 2 kWh. In terms of load power        deviation, HEMS 506 prefers to discharge (charge) its ESS, if        extant, at ω₁ (ω₂). If the rating power of the ESS is 1 kW for        both charging and discharging, the HEMS real-time consumptions        are 3 kW and 1 kW at ω₁ and ω₂, respectively.    -   Since uncontrollable load 507 has a fixed demand of 10 kW, the        total real-time loads of the isolated system are 13 kW and 11 kW        at ω₁ and ω₂, respectively. The PV generation may fluctuate in        real-time, e.g., 12.5 kW at ω₁ and 11.5 kW at ω₂. To maintain        real-time balance, utility-scale PV installation 505 needs to        deploy its ancillary ESS as described previously (Algorithm 1,        operation 7). If the rating power of the ESS is also 1 kW, the        real-time balancing model will request the ESS to discharge        (charge) 0.5 kW at ω₁(ω₂). Load shedding or PV curtailment may        take place if the ESS cannot fully compensate for the real-time        fluctuations.    -   When τ₁ has passed, the entire procedure will repeat (Algorithm        1, operation 8) to solve for τ₂.

The techniques described herein were tested on a modified IEEE 123-bustest feeder and a real-world unbalanced utility system with 3994 buses.The results of such testing are provided below.

FIG. 6 is a graphical diagram illustrating a modified IEEE 123-bus testfeeder (system 602) that was employed for testing power systemrestoration incorporating diverse distributed energy resources, inaccordance with one or more aspects of the present disclosure. As shownin the example of FIG. 6, system 602 has 123 buses and 126 lines. It wasassumed that all three-phase lines and the added four tie switches(indicated by the bolder lines and dashed lines, respectively, in FIG.6) are switchable. System 602 consists of 200 residential houses and 7utility-scale DERs. Key parameters of utility-scale DERs and thebehind-the-meter DERs in residential houses are summarized in Table I.In this test case, it was assumed that each residential house is managedby a unique HEMS, and each household has a heat pump, a water heater,and an ESS, as listed in Table I. In addition, the household PVpenetration level was set to 50%. That is, 100 houses were randomlyselected and designated to have installed rooftop PV panels. Moreover,among these selected 100 houses, their rooftop PV capacities were alsorandomly selected from the three following values: 4 kW, 6 kW, and 8 kW.

TABLE I STATISTICS OF THE SIMULATED TEST SCENARIOS Utility-scale DERdata Regulation ESS Capacity capacity capacity Type Bus (MVA) (kW) (kWh)Substation 149 5.0 N/A N/A Diesel 21, 64, 108 1.0 10.0 N/A PV 35, 78 0.66.0 10.0 Wind 48, 95 0.8 8.0 10.0 Behind-the-meter DER data RatingCapacity power Comfort index: Type (kW) (kWh) comfort range ESS 5.0 2.5State of charge: [10%, 90%] Heat N/A 3.0 Indoor temperature: pump [20°C., 24° C.] Water N/A 4.0 Water temperature: heater [45° C., 50°C.]

The MPC horizon and MPC interval (ΔT) of adaptive restoration planningwere 6 hours and 15 minutes, respectively. The real-time interval (Δt)was 1 minute. In the distribution restoration stage, the majoruncertainties come from renewable DER generation and load variation.When solving the MPC-based planning study, it was assumed the currentMPC interval has no forecast error, while the forecast errors in theforeseen MPC intervals follows Gaussian distribution N(0, 2.5%).Although the forecast error of the current MPC interval was ignored,real-time fluctuations still existed. It was also assumed that thereal-time deviation from the MPC forecast follows Gaussian distributionN(0, 1.5%).

The distribution system line outage sequences throughout the 6-hourplanning period are described in Table II, and the time is given inhh:mm format. In some studies in the literature, the repair coordinationis investigated and the repair sequence is treated as a variable.However, the repair coordination is outside of the scope of thisdisclosure, and thus the repair sequences are given as parameters inTable II. If a repair time is not given in Table II, e.g., line 30-250,the corresponding line was assumed to remain out-of-service at the endof the studied 6-hour horizon. It was also assumed that the substation(Bus 149) has no access to power supply because of outages in theupstream power systems. Note that the DSO was assumed to be not able toforesee future failures and thus only relied on current faultinformation when optimizing the restoration plan. The proposed fouroptimization models can be categorized as either linear or quadraticprogramming models, and they were solved using Cplex in the GeneralAlgebraic Modeling System from GAMS Development Corp. (GAMS/CPLEX) on alaptop computer with a quad-core i7 processor from Intel Corp. and 16-GBRAM.

TABLE II LINE OUTAGE SEQUENCES OF THE SIMULATED TEST SCENARIOS Repairtime Line/Switch Outage time (if applicable) 13-18 T_(st) + 00:00T_(st) + 05:15  30-250 T_(st) + 01:45 N/A 50-51 T_(st) + 03:30 N/A 86-87T_(st) + 00:00 T_(st) + 01:15 105-108 T_(st) + 00:00 N/A  52-152T_(st) + 00:00 T_(st) + 04:15  67-160 T_(st) + 00:00 N/A

To evaluate the impacts of intermittent renewable DERs (especially solarPV), two test cases were simulated with the same fault settings shown inTable II except for the outage occurrence time (i.e., T_(st)):

-   -   Case 1: T_(st)=6 a.m., the PV generation has an increasing trend        for the 6-hour MPC horizon.    -   Case 2: T_(st)=12 p.m., the PV generation has a decreasing trend        for the 6-hour MPC horizon.

The linearized unbalanced power flow solution was compared with the fullAC power flow benchmark results from OpenDSS distributed by the ElectricPower Research Institute. The normalized root-mean-squared error ofvoltage magnitude, nodal active power injection, and nodal reactivepower injection were 0.14%, 0.61%, and 1.54%, respectively. Thus, it wasconcluded that the employed power flow linearization technique canprovide a very good approximation.

According to the computational time calculation discussed above, theaverage computational time for MPC-based restoration planning andreal-time dispatch were 1.46 seconds and 0.95 seconds, respectively.Referring to the 15-min MPC interval and 1-min real-time interval, thetechniques described herein were able to successfully meet thecomputational requirement for post-event restoration decision-making andreal-time control.

FIGS. 7A and 7B are conceptual diagrams illustrating switch operationsduring two test cases of power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure. The switch positions are dynamically updated inaccordance with the techniques described herein to accommodatefluctuations in renewable generation and load consumption. As shown inFIGS. 7A and 7B, the operation sequences of switch 91-93 are the same inboth cases. Different switches are opened/closed during the first 2hours of the outage in these two cases, mainly because of the differentrenewable DER generation profiles.

The performance of these two test cases are summarized in Table III.

TABLE III STATISTICS OF THE SIMULATED TEST CASES Case 1 Case 2 MPC-basedrestoration planning Total restored load 15.05 14.93 consumption (MWh)Total renewable DER 8.28 7.58 generation (MWh) Total uncontrollable load14.04 13.96 consumption (MWh) Net energy consumption of 1.01 0.97 HEMS(MWh) Peak/valley consumption of  0.42/−0.03  0.30/−0.03 HEMS (MW) Totalload energy not 2.49 2.65 supplied (MWh) Maximum/minimum load 0.73/0.00.78/0.20 shedding (MW) Maximum/minimum voltage 1.012/0.985 1.016/0.986magnitude (p.u.) Real-time dispatch Net energy consumption 0.91 0.88 ofHEMS (MWh) Peak/valley consumption  0.41/−0.03  0.30/−0.11 of HEMS (MW)Total load energy not supplied 2.89 3.23 Total renewable energy 3.193.73 curtailment (kWh) Maximum/minimum voltage 1.015/0.985 1.019/0.988magnitude (p.u.)

In both cases, load shedding was inevitable because of the limitedgenerating capacity and intermittency of renewable DERs. The unsuppliedload energy in Case 2 was slightly higher than that of Case 1 because ofthe reduction in renewable DER output. Several interesting observationscan be found by comparing the MPC-based planning results with thereal-time dispatch results. First, the real-time net HEMS energyconsumptions were smaller than the MPC-based planning solutions in bothcases, indicating that the HEMS might discharge their ESS morefrequently and reduce their comfort settings in real-time to follow theMPC-based planning reference. This is also supported by the comparisonof peak/valley HEMS consumption in both simulated cases. Second, theconsideration of real-time forecast errors and intermittency resulted inhigher load shedding in both cases. This also validates the necessity tointegrate real-time balancing into restoration planning. Note that ineither case, the fluctuating renewable DER generation will lead to avery small amount of curtailment because of the scarcity of flexibleresources. In general, the real-time dispatch solutions do not deviatemuch from the MPC-based planning solutions, as shown in Table III.

FIGS. 8A-8L are graphical plots illustrating detailed results of a testcase (Case 1) of power system restoration incorporating diversedistributed energy resources, in accordance with one or more aspects ofthe present disclosure. Specifically, FIG. 8A shows the total loadrestoration profile, with line 802 providing the real-time results andline 803 providing the MPC results. FIG. 8B shows the utility PVgeneration at Bus 78, with line 804 providing the MPC forecast when theoutage occurs, line 805 representing actual PV generation (15-minslots), and line 806 representing real-time PV generation (1-min slots).FIG. 8C shows ESS operation at Bus 78, with line 807 representing thebattery power output and line 808 representing the batterystate-of-charge. FIG. 8D shows the schedule of HEMS-1, with line 809representing the MPC results, and line 810 representing the real-timeresults. FIG. 8E shows the behind-the-meter DER operations of HEMS-1,with line 811 representing the water heater and line 812 representingthe heat pump. FIG. 8F shows ESS operation at HEMS-1, with line 813representing the battery power output and line 814 representing thebattery state-of-charge. FIG. 8G shows the schedule of HEMS-2, with line815 representing the MPC results and line 816 representing the real-timeresults. FIG. 8H shows the behind-the-meter DER operations of HEMS-2,with line 817 representing the water heater and line 818 representingthe heat pump. FIG. 8I shows ESS operation at HEMS-2, with line 819representing the battery power output, line 820 representing the batterystate-of-charge, and line 821 representing the rooftop PV generation.FIG. 8J shows the schedule of HEMS-3, with line 822 representing the MPCresults and line 823 representing the real-time results. FIG. 8K showsthe behind-the-meter DER operations of HEMS-3, with line 824representing the water heater and line 825 representing the heat pump.Finally, FIG. 8L shows PV/ESS operation at HEMS-3, with line 826representing the battery power output, line 827 representing the rooftopPV generation, and line 828 representing the battery state-of-charge.

As seen in FIGS. 8A and 8B, the restored load power and utility PVgeneration share similar increasing trends. As seen in FIG. 8A, thevolatility of the restored load consumption does not become significantuntil approximately 3.5 hours after the occurrence of the outage. Thisphenomenon is caused by the exhaust of flexible resources, such as ESS.FIG. 8C shows the real-time operation schedule and state-of-chargeprofile of the ESS associated with the utility-scale PV. The ESS stateof charge is kept near 50% right after the outage occurs, but it dropsdramatically after 3 hours of operation; thus, the ESS's capability tocompensate for the variable renewable generation is greatly restrainedin the second half of the MPC horizon.

The real-time HEMS energy consumption was also analyzed. FIGS. 8D-8L maybe used to compare the following three HEMSs:

-   -   HEMS-1: connected to bus 41, phase C, and did not have a rooftop        PV system installed;    -   HEMS-2: connected to bus 41, phase C, and was equipped with a 4        kW rooftop PV system;    -   HEMS-3: connected to bus 85, phase C, and was equipped with a 4        kW rooftop PV system.

As shown in FIG. 8D, the real-time consumption of HEMS-1 almostperfectly follows the MPC reference. In contrast, FIGS. 8G and 8J showthat the real-time consumptions of HEMS-2 and HEMS-3 are much morevolatile. The reason lies in the variable generation output of therooftop PV panels of HEMS-2 and HEMS-3. The exhaustion of household ESScapability also contributes to deviations in real-time. As seen in FIG.8I, the ESS of HEMS-2 is operating with its lowest state of charge after2 hours of outage, which further leads to a fluctuating real-time loadschedule at that period. Similarly, the ESS state-of-charge of HEMS-3remains close to its lower bound after 3 hours of outage as shown inFIG. 8L, resulting in a volatile real-time consumption in FIG. 8J. Incomparison, the state-of-charge of the ESS of HEMS-1 does not reach itslimit throughout the 6-hour MPC horizon, which contributes to the smoothreal-time consumption shown in FIG. 8D. FIGS. 8E, 8H, and 8K demonstratethe real-time schedules of water heaters and heat pumps and verify theeffectiveness of the real-time behind-the-meter DER models with discreteon/off control signal.

HEMS-1 and HEMS-2 are located at the same bus and phase. The differencesin FIGS. 8D-8F and FIGS. 8G-8I are mainly caused by the rooftop PV ofHEMS-2. On the other hand, the major difference between HEMS-2 andHEMS-3 is the location. As the distribution system may be divided intoseveral isolated subnetworks due to faults, the location of a HEMS playsan important role in determining the availability of generatingresources. For example, the generations from rooftop PVs are increasingin the simulated 6-hour horizon as observed from FIGS. 8I and 8L.According to FIG. 8I, HEMS-2 is capable of charging its ESS toapproximately 50% state-of-charge by the end of simulation using theabundant PV generation. However, the ESS state-of-charge of HEMS-3remains low despite the PV generation being high, possibly because thegeneration capacity of the island where HEMS-3 is located isinsufficient, so rooftop PV generations are required to supply othercritical loads.

Using Case 2 as a benchmark, the techniques of the present disclosurewere compared with several commonly employed restoration strategies tofurther validate effectiveness. The same parameter settings used in thebenchmark case were adopted in the comparative study. The comparativecases are briefly described as follows:

-   -   Case 3: Static restoration planning based on forecast data.        There is no dynamic update or real-time dispatch.    -   Case 4: The proposed MPC-based adaptive restoration planning.        There is no real-time dispatch coordination.    -   Case 5: Static restoration planning as in Case 3 with        collaborative real-time dispatch.    -   Case 6: The techniques described herein, but the restoration        planning solution is derived while ignoring the phase unbalances        (i.e., using balanced power flow in operation 3 of Algorithm1).

The comparison results are listed in Table IV.

TABLE IV PERFORMANCES OF COMPARATIVE CASES AGAINST BENCHMARK CASE 2Total load energy Peak curtailed Total utility-scale PV and not supplied(%) load power (%) wind generation (%) Case 2 100.0 100.0 100.0 Case 3201.10 172.41 92.55 Case 4 147.22 135.02 96.89 Case 5 175.97 131.3295.54 Case 6 188.85 181.39 87.45

The statistics in Table IV indicate the relative ratio in comparisonwith the benchmark results (Case 2). For example, 201.10%, as shown inthe third row, the second column, denotes that compared to Case 2, Case3 curtailed 201.10% of the total curtailed load energy. Note that Case 3and Case 4 do not include real-time dispatch, so it is assumed that theHEMS will randomly allocate the consumption schedules ofbehind-the-meter DERs in real-time to cope with the restoration planningsolution. The following conclusions can be observed from Table IV:

-   -   (1) Case 3 has the worst performance among all cases. The        forecast errors cannot be mitigated and network topology is        static because of the absence of MPC-based rolling optimization.        Additionally, the arbitrary real-time behaviors of        behind-the-meter DERs contributes to the significantly higher        load curtailment.    -   (2) The performance of Case 4 is much better than Case 3,        indicating the effectiveness of the MPC-based adaptive        restoration planning strategy disclosed herein; however, the        results of Case 4 are still worse than the benchmark Case 2        because the real-time fluctuations have not been handled.    -   (3) Compared to Case 3, the introduction of real-time dispatch        to Case 5 results in performance improvement. On the other hand,        Case 5 still has higher load curtailment than Case 4 because        real-time fluctuations are not as significant as long-term        forecast errors. Note that Case 5 has the lowest peak load        curtailment power. This is because proper real-time        behind-the-meter DER scheduling can greatly level off load        consumption spikes, leading to a smaller load curtailment.    -   (4) Distribution systems are inherently unbalanced. In Case 6,        the phase unbalances are ignored when reconfiguring the        distribution grid. As shown in Table IV, Case 6 has more        unsupplied load than Case 2. This can be explained by the        following two aspects: (i) the balanced power flow employed        inaccurate network data, thus the solution contains considerable        errors; and (ii) the restoration solution based on balanced        power flow overestimates the available generation capacity, thus        could become infeasible for real-time operation. For example,        the balanced power flow may require a rooftop PV to supply        another load at a different phase, which is technically        infeasible and will possibly result in additional load shedding        and renewable curtailment in real-time.

The results in Table IV demonstrate that either the MPC-basedrestoration planning or the real-time dispatch cannot fully manage theuncertainty and intermittency associated with renewable DERs andbehind-the-meter DERs. The efficiency and necessity of the techniquesdescribed herein to coordinate system-level restoration andbehind-the-meter DER control are validated. Moreover, the necessity toemploy unbalanced power flow for distribution system restoration studiesis validated.

A real-world unbalanced utility system was also employed to validate theperformance of the techniques of the present disclosure on largedistribution systems. The test system is a radial system serving aresidential area with 3994 buses, 9 utility-scale PVs (each of them hasa capacity of 200 kW), and 163 controllable loads. It was assumed thateach controllable load is managed by a HEMS. The parameters of utilityPVs, behind-the-meter DERs, and simulation horizon/interval were thesame used in the IEEE 123 bus system. To simulate the power outagescenario, in this test case the substation did not provide power supplythroughout the entire simulation period, and 10 randomly selecteddistribution lines are disconnected. The outage was assumed to start atT_(st)=9 a.m.

For the 3994-bus system, the average computational time for MPC-basedrestoration planning and real-time dispatch were 26.76 seconds and 1.55seconds, respectively. Although it took much longer to solve the3994-bus system than the IEEE 123 bus system because of the increasedsize, it still fits in the 15-min MPC interval. Thus, the techniquesdescribed herein can be successfully applied to real-world largedistribution systems as well.

Compared with the OpenDSS benchmark results, the normalizedroot-mean-squared error of voltage magnitude, nodal active powerinjection, and nodal reactive power injection were 0.21%, 1.14%, and3.78%, respectively. Therefore, the error introduced by power flowlinearization is not significant.

To validate the techniques for power system restoration incorporatingdiverse distributed energy resources disclosed herein, the following twocases were compared on this utility test system:

-   -   Case 7: The techniques of the present disclosure.    -   Case 8: The techniques of the present disclosure without the        DSO-HEMS collaborations.

That is, the behind-the-meter DERs are treated as uncontrollable loads.

The comparison of these two cases are listed in Table V where theresults of Case 7 serve as the benchmark.

TABLE V PERFORMANCES COMPARISON ON THE LARGE UTILITY TEST SYSTEM Totalload energy Peak curtailed Utility-scale PV not supplied (%) load power(%) curtailment (%) Case 7 100.0 100.0 100.0 Case 8 129.81 145.63 193.42

It is observed that the absence of behind-the-meter DER collaborationwill lead to a significant increase in load shedding and renewableenergy curtailment. Especially in this utility system, where allgeneration resources are PVs, the flexibility of behind-the-meter DERsacts as a critical buffer to level off the volatility of PVs andenhances the reliability of power supply during an outage.

FIG. 9 is a flow diagram illustrating example operations for performingpower system restoration incorporating diverse distributed energyresources, in accordance with one or more aspects of the presentdisclosure. FIG. 9 represents only one example process for performingpower system restoration incorporating diverse distributed energyresources as described herein, and various other or additionaloperations may be used in other examples. The example operations of FIG.9 are described below within the context of FIG. 1.

In the example of FIG. 9, an aggregation unit may be configured toreceive a service collaboration request indicating a problem in a powerdistribution network (900). For example, aggregation unit 10B mayreceive a service collaboration request from power management unit 4.

Responsive to receiving the service collaboration request, theaggregation unit may, in the example of FIG. 9, determine an optimizedoperation schedule covering a model predictive control (MPC) horizonduration for one or more DERs in the power distribution network undermanagement by the aggregation unit (902). For example, aggregation unit10B may determine an optimized operation schedule that includesaggregator-managed DERs 8B, 8C, and 8D. The optimized operation schedulemay be determined based on respective minimum and maximum real andreactive power values for the one or more DERs.

In the example of FIG. 9, the aggregation unit may determine, based onthe optimized operation schedule, an estimated flexibility range fordevices under management by the aggregation unit (904). The aggregationunit may output an indication of the estimated flexibility range (906).For example, aggregation unit 10B may determine an estimated flexibilityrange for devices including aggregator-managed DERs 8B, 8C, and 8D.Aggregation unit 10B may output estimated flexibility range 12B to oneor more other components of system 2 via one or more wired or wirelesscommunication networks.

In the example of FIG. 9, a power management unit may be configured toreceive the indication of the estimated flexibility range (908). Basedon a linearized restoration model of the three-phase unbalanced powerdistribution network that includes the indication of the estimatedflexibility range, the power management unit may determine an optimizedpower distribution network reconfiguration plan and an overall optimizedoperation schedule covering the MPC horizon duration for both energyresources under management by the power management unit and the one ormore DERs (910). For example, power management unit 4 may receiveestimated flexibility range 12B and estimated flexibility range 12Aoutput by aggregation unit 10A. Power management unit 4 may incorporateestimated flexibility ranges 12A and 12B into a model of the powerdistribution network and determine an optimized power distributionnetwork reconfiguration plan and an overall optimized operation schedulefor the network.

In the example of FIG. 9, the power management unit may cause, based onthe optimized power distribution network reconfiguration plan, areconfiguration of the power distribution network (912). As one example,power management unit 4 may output switch instruction 13 for use bysystem reconfiguration device 6B. As another example, power managementunit 4 may output the optimized power distribution networkreconfiguration plan for use by one or more other components of thepower distribution network, such as a distribution system manager.

The power management unit may output an indication of the overalloptimized operation schedule (914). For instance, power management unit4 may output optimized schedule 14 to one or more other components ofsystem 2 via one or more wired or wireless communication networks.

In the example of FIG. 9, the power management unit may cause one ormore of the energy resources under management by the power managementunit to modify operation based on the overall optimized operationschedule (916). As one example, power management unit 4 may outputsetpoints 15 for use by PMU-managed DER 9. As another example, powermanagement unit 4 may output an indication of the overall optimizedoperation schedule to one or more other components of system 2 for usein determining setpoints to be used by PMU-managed DER 9.

The aggregation unit may, in the example of FIG. 9, be furtherconfigured to receive the indication of the overall optimized operationschedule (918). Based on the indication of the overall optimizedoperation schedule, the aggregation unit may determine setpoints for theone or more DERs (920). For example, aggregation unit 10B may receiveoptimized schedule 14 and use it to determine setpoints foraggregator-managed DERs 8B, 8C, and 8D.

The aggregation unit may, in the example of FIG. 9, cause at least oneof the one or more DERs to modify operation based on the setpoints(922). For example, aggregation unit 10B may send instructions toaggregator-managed DERs 8B, 8C, and/or 8D instructing them to operate atthe provided setpoints. As another example, aggregation unit 10B mayoutput the setpoints to one or more other components (e.g., an inverter)that may instruct an aggregator managed DER to use the receivedsetpoints.

The example operations of FIG. 9 may be performed in an iterativefashion. That is, while only a single flow is shown, each of operations900, 902, 904, 906, 908, 910, 912, 914, 916, 918, 920, and/or 922 may beperformed any number of times. In some examples, the operations areperformed in an iterative fashion. In some such examples, the frequencywith which these operations are performed may be the same. In other suchexamples, one or more of the operations may be performed with higher orlower frequency than other operations. For instance, operations 902,904, 906, 908, 910, 912, 914, and 918 may be performed at a firstfrequency that represents an MPC interval. Operations 916, 920, and 922may be performed at a second, higher frequency that represents areal-time interval.

The techniques disclosed herein provide a collaborative framework fordistribution system restoration that coordinates the adaptiverestoration planning and real-time dispatch through utility-scale andaggregator-controlled DERs, including behind-the-meter DERs. Theflexibility of such DERs is quantified by aggregators and used by DSOsduring the restoration planning stage. The MPC technique is employedherein to dynamically update the system restoration strategy toaccommodate the up-to-date renewable generation forecast, the faultpropagation, and the flexibility range of aggregator-controlled DERs. Toaccommodate fluctuations in the real-time domain, real-time models forDSOs and aggregator-controlled DERs are provided to maintain powerbalance based on the reference obtained in the MPC-based restorationplanning. The performance of the techniques disclosed herein are alsoverified in case studies. Compared to existing restoration planningstudies, the techniques of the present disclosure are effective inhandling uncertainty factors, accommodating flexibility ofaggregator-controlled DERs, and dynamically adjusting restorationschemes during the restoration process.

At least some of the techniques of the present disclosure may beadditionally or alternatively described by one or more of the followingexamples.

Example 1. A system comprising: an aggregation unit comprising at leastone processor, the aggregation unit being configured to: receive aservice collaboration request indicating a problem in a powerdistribution network; responsive to receiving the service collaborationrequest, determine, based on respective minimum and maximum real andreactive power values for one or more DERs in the power distributionnetwork under management by the aggregation unit, an optimized operationschedule covering a model predictive control (MPC) horizon duration forthe one or more DERs; determine, based on the optimized operationschedule, an estimated flexibility range for devices under management bythe aggregation unit; and output an indication of the estimatedflexibility range; a power management unit comprising at least oneprocessor, the power management unit being configured to: receive theindication of the estimated flexibility range; determine, based on alinearized restoration model of the three-phase unbalanced powerdistribution network that includes the indication of the estimatedflexibility range, an optimized power distribution networkreconfiguration plan and an overall optimized operation schedulecovering the MPC horizon duration for both energy resources undermanagement by the power management unit and the one or more DERs; cause,based on the optimized power distribution network reconfiguration plan,a reconfiguration of the power distribution network; output anindication of the overall optimized operation schedule; and cause one ormore of the energy resources under management by the power managementunit to modify operation based on the overall optimized operationschedule, wherein the aggregation unit is further configured to: receivethe indication of the overall optimized operation schedule; determine,based on the indication of the overall optimized operation schedule,setpoints for the one or more DERs; and cause at least one of the one ormore DERs to modify operation based on the setpoints.

Example 2. The system of example 1, wherein: the aggregation unit isfurther configured to determine the optimized operation schedule,determine the estimated flexibility range, and output the indication ofthe estimated flexibility range iteratively at a first frequency thatrepresents an MPC interval, the power management unit is furtherconfigured to receive the indication of the estimated flexibility range,determine the optimized power distribution network reconfiguration planand the overall optimized operation schedule, cause the reconfigurationof the power distribution network, and output the indication of theoverall optimized operation schedule iteratively at the first frequency,the aggregation unit is further configured to determine the setpointsfor the one or more DERs and cause the at least one of the one or moreDERs to modify operation iteratively at a second frequency thatrepresents a real-time interval, the power management unit is furtherconfigured to cause the one or more of the energy resources undermanagement by the power management unit to modify operation iterativelyat the second frequency, and the MPC interval is larger than thereal-time interval.

Example 3. The system of any of examples 1-2, wherein: the aggregationunit is further configured to output an indication of the optimizedoperation schedule, and the power management unit is configured todetermine the optimized power distribution network reconfiguration planand the overall optimized operation schedule based further on theoptimized operation schedule.

Example 4. The system of any of examples 1-3, wherein the aggregationunit is configured to cause at least one of the one or more DERs tomodify operation based on the setpoints by outputting, for use by the atleast one of the one or more DERs, an indication of the setpoints.

Example 5. The system of any of examples 1-4, wherein the powermanagement unit is configured to determine the overall optimizedoperation schedule covering the MPC horizon duration for both energyresources under management by the power management unit and the one ormore DERs by determining: Min Σ_(τ=T) _(st) ^(T) ^(en) Σ_(n∈Ω) _(N)[γ_(n) ^(G)P_(n,τ) ^(G)+γ_(n) ^(S)Σ_(ϕ∈Ψ) _(n) (P_(n,τ) ^(S,ϕ)+Q_(n,τ)^(S,ϕ))]+Σ_(τ=T) _(st) ^(T) ^(en) Σ_(g∈Ω) _(H) (γ_(h) ^(V)P_(h,τ)^(V)+γ_(h) ^(W)P_(h,τ) ^(W)), subject to: P_(n,τ) ^(G,min)≤P_(n,τ)^(G)≤P_(nτ) ^(G,max), Q_(n,τ) ^(G,min)≤Q_(n,τ) ^(G)≤Q_(nτ) ^(G,max),P_(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(nP) _(n,τ) ^(G,ϕ), Q_(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(nQ)_(n,τ) ^(G,ϕ), P_(n,τ) ^(G,ϕ)−P_(n,τ) ^(D,ϕ)+P_(n,τ) ^(S,ϕ)=Σ_(nm∈Ω)_(L) (P_(nm,τ) ^(ϕ)−_(mn,τ) ^(ϕ)), Q_(n,τ) ^(G,ϕ)−Q_(n,τ) ^(D,ϕ)+Q_(n,τ)^(S,ϕ)=Σ_(nm∈Ω) _(L) (Q_(nm,τ) ^(ϕ)−Q_(mn,τ) ^(ϕ)), P_(n,τ)^(D,ϕ)=P_(n,τ) ^(D,fix,ϕ)+Σ_(h→n,ϕ))P_(h,τ), Q_(n,τ) ^(D,ϕ)=Q_(n,τ)^(D,fix,ϕ)+Σ_(h→(n,ϕ))Q_(h,τ), 0≤P_(n,τ) ^(S,ϕ)≤P_(n,τ) ^(D,ϕ),0≤Q_(n,τ) ^(S,ϕ)≤Q_(n,τ) ^(D,ϕ), V_(n,τ) ^(ϕ)−V_(m,τ) ^(ϕ)=2(r_(nm)^(ϕϕ)P_(nm,τ) ^(ϕ)+x_(nm) ^(ϕϕ)Q_(nm,τ) ^(ϕ))−2ϑ₁(r_(nm) ^(ϕϕ′)P_(nm,τ)^(ϕ′)+r_(nm) ^(ϕϕ″)P_(nm,τ) ^(ϕ″)+x_(nm) ^(ϕϕ′)Q_(nm,τ) ^(ϕ′)+x_(nm)^(ϕϕ″)Q_(nm,τ) ^(ϕ″))−2ϑ₂(x_(nm) ^(ϕϕ′)P_(nm,τ) ^(ϕ′)−x_(nm)^(ϕϕ″)P_(nm,τ) ^(ϕ″)−r_(nm) ^(ϕϕ′)Q_(nm,τ) ^(ϕ′)+r_(nm) ^(ϕϕ″)Q_(nm,τ)^(ϕ″)), V_(n) ^(ϕ,min)≤V_(n,τ) ^(ϕ)≤V_(n) ^(ϕ,max), P_(h,τ)^(W)≥P_(h,τ)−p_(h,τ)*, P_(h,τ) ^(W)≥p_(h,τ)*−P_(h,τ), p_(h,τ)^(min)≤P_(h,τ)≤p_(h,τ) ^(max), e_(h,τ) ^(min)−P_(h,τ) ^(V)≤Σ_(T=T) _(st)^(τ)P_(h,T)≤e_(h,τ) ^(max)P_(h,τ) ^(V), and P_(h,τ) ^(V)≥0, wherein: τrepresents an MPC interval, T_(st) and T_(en) represent a start time andan end time, respectively, of the MPC horizon, n represents a bus in thepower distribution network, Ω_(N) represents a set of buses in the powerdistribution network, γ_(n) ^(G) represents a cost of energy generationby the energy resources under management by the power management unit atbus n, P_(n,τ) ^(G) and Q_(n,τ) ^(G), represent active and reactivepower generation, respectively, by the energy resources under managementby the power management unit at bus n during MPC interval τ, γ_(n) ^(S)represents a penalty of load shed at bus n, ϕ represents a phase inΨ_(n), Ψ_(n) represents a set of phases at bus n, P_(n,τ) ^(S,ϕ) andQ_(n,τ) ^(S,ϕ) represent active and reactive power generation,respectively, by the energy resources under management by the powermanagement unit on phase ϕ at bus n that is shed during MPC interval τ,h represents an aggregation unit in the power distribution network,Ω_(H) represents a set of aggregation units in the power distributionnetwork, γ_(h) ^(V) represents a penalty of aggregation unit h violatingits estimated flexibility range, P_(n,τ) ^(V) represents a violation ofaggregation unit's h estimated flexibility range in MPC interval τ,γ_(h) ^(W) represents a penalty of aggregation unit h violating itsoptimized operation schedule, P_(n,τ) ^(W) represents a deviation ofaggregation unit h from its optimized operation schedule in MPC intervalτ, P_(n,τ) ^(G,min) and P_(n,τ) ^(G,max) represent a minimum and maximumactive power generation, respectively, by the energy resources undermanagement by the power management unit at bus n during MPC interval τ,Q_(n,τ) ^(G,min) and Q_(n,τ) ^(G,max) represent a minimum and maximumreactive power generation, respectively, by the energy resources undermanagement by the power management unit at bus n during MPC interval τ,P_(n,τ) ^(G,ϕ) and Q_(n,τ) ^(G,ϕ) represent active and reactive powergeneration, respectively, by the energy resources under management bythe power management unit on phase (at bus n during MPC interval τ,P_(n,τ) ^(D,ϕ), and Q_(n,τ) ^(D,ϕ) represent active and reactive loadconsumption, respectively, by phase ϕ at bus n during MPC interval τ, nmrepresents a distribution line from bus n to bus m in the powerdistribution network, Ω_(L) represents a set of distribution lines inthe power distribution network, P_(nm,τ) ^(ϕ) and Q_(nm,τ) ^(ϕ)represent the active and reactive power flow, respectively, through linenm during MPC interval τ, P_(mn,τ) ^(ϕ) and Q_(mn,τ) ^(ϕ) represent theactive and reactive power flow, respectively, through line mn during MPCinterval τ, P_(n,τ) ^(D,fix,ϕ) and Q_(n,τ) ^(D,fix,ϕ) represent activeand reactive uncontrollable load consumption, respectively, by phase (atbus n during MPC interval τ, P_(h,τ) and Q_(h,τ) represent optimizedactive and reactive loads, respectively, for aggregation unit h duringMPC interval τ, V_(n,τ) ^(ϕ) and V_(m,τ) ^(ϕ) represent a squared nodalvoltage magnitude on phase ϕ at bus n and bus m, respectively, r_(nm)^(ϕϕ) and x_(nm) ^(ϕϕ) represent the resistance and reactance,respectively, of line nm, ϕ′ and ϕ″ represent the phase that leads andlags ϕ, respectively, by 2π/3, ϑ₁ and ϑ₂ represent cos (2π/3) and sin(2π/3), respectively, V_(n) ^(ϕ,min) and V_(n) ^(ϕ,max) represent aminimum and maximum value of V_(n) ^(ϕ), respectively, P_(h,τ)*represents unit h's optimized operation schedule in MPC interval τ,P_(h,τ) ^(min) and P_(n,τ) ^(max) represent a minimum and maximum activepower, respectively, for aggregation unit h during MPC interval τ, ande_(h,τ) ^(min) and e_(h,τ) ^(max) a minimum and maximum energyconsumption, respectively, for aggregation unit h during MPC interval τ.

Example 6. The system of any of examples 1-5, wherein the aggregationunit is configured to determine the optimized operation schedulecovering the MPC horizon for the one or more DERS by determining:

MinΣ_(τ = T_(st))^(T_(en))(Σ_(a ∈ Ω_(A_(h)))p_(h, τ)^(a) + Σ_(b ∈ Ω_(B_(h)))β_(h)^(b)s_(h, τ)^(b) − s_(h)^(b, pre)₂²)

subject to: p_(h,τ) ^(a,min)≤p_(h,τ) ^(a)≤p_(h,τ) ^(a,max), q_(h,τ)^(a,min)≤q_(h,τ) ^(a)≤q_(h,τ) ^(a,max), s_(h,τ) ^(b,min)≤s_(h,τ)^(b)≤s_(h,τ) ^(b,max), s_(h,τ) ^(b)=s_(h,τ−1) ^(b)+Σ_(a→b)δ_(h)^(a)p_(h,τ) ^(a)ΔT+c_(h,τ) ^(b)+ΔT, wherein: τ represents an MPCinterval, T_(st) and T_(en) represent a start time and an end time,respectively, of the MPC horizon, Ω_(A) _(h) represents the one or moreDERs, a represents a DER in the one or more DERs, p_(h,τ) ^(a)represents an active power of DER a for MPC interval τ, Ω_(B) _(h)represents a set of comfort parameters, b represents a comfort parameterin the set of comfort parameters, s_(h,τ) ^(b) represents a value ofcomfort parameter b for MPC interval τ, s_(h,τ) ^(b,pre) represents apreferred value of comfort parameter b for MPC interval τ, p_(h,τ)^(a,min) and p_(h,τ) ^(a,max) represent a minimum and maximum activepower of DER a, respectively, for MPC interval τ, s_(h,τ) ^(b,min) ands_(h,τ) ^(b,max) represent a desired minimum and maximum value,respectively, of comfort parameter b for MPC interval τ, s_(h,τ−1) ^(b)represents the value of comfort parameter b for MPC interval τ−1, a→bindicates that power consumption by DER a will influence comfortparameter b, δ_(h) ^(a) represents a DER conversion efficiency factorfor DER a, ΔT represents the duration of MPC interval τ, and c_(h,τ)^(b) represents an external influence factor on comfort parameter b forMPC interval τ.

Example 7. The system of any of examples 1-6, wherein the aggregationunit is configured to determine the estimated flexibility range bydetermining:

p_(h, τ)^(min) = Σ_(a ∈ Ω_(A_(h)))p_(h, τ)^(a, min ), p_(h, τ)^(max) = ∑_(a ∈ Ω_(A_(h)))p_(h, τ)^(a, max ), and  ${{Min}\mspace{14mu} e_{h,\tau}^{\min}} = {\sum_{T = T_{st}}^{\tau}{\sum_{a \in \Omega_{A_{h}}}{p_{h,T}^{a}\mspace{14mu}{and}}}}$ Max  e_(h, τ)^(max) = Σ_(T = T_(st))^(τ)Σ_(a ∈ Ω_(A_(h)))p_(h, T)^(a),

each subject to: p_(h,τ) ^(a,min)≤p_(h,τ) ^(a)≤p_(h,τ) ^(a,max), q_(h,τ)^(a,min)≤q_(h,τ) ^(a)≤q_(h,τ) ^(a,max), s_(h,τ) ^(b,min)≤s_(h,τ)^(b)≤s_(h,τ) ^(b,max), and s_(h,τ) ^(b)=s_(h,τ−1) ^(b)+Σ_(a→b)δ_(h)^(a)p_(h,τ) ^(a)ΔT+c_(h,τ) ^(b)+ΔT, wherein: i represents an MPCinterval, T_(st) represents a start time of the MPC horizon, e_(h,τ)^(min) and e_(h,τ) ^(max) represent a minimum and maximum energyconsumption of devices under management by the aggregation unit for MPCinterval τ, Ω_(A) _(h) represents the one or more DERs, a represents aDER in the one or more DERs, p_(h,τ) ^(a) represents an active power ofDER a for MPC interval τ, b represents a comfort parameter in a set ofcomfort parameters, s_(h,τ) ^(b) represents a value of comfort parameterb for MPC interval τ, s_(h,τ) ^(b,pre) represents a preferred value ofcomfort parameter b for MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ)^(a,max) represent a minimum and maximum active power of DER a,respectively, for MPC interval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max)represent a desired minimum and maximum value, respectively, of comfortparameter b for MPC interval τ, s_(h,τ−1) ^(b) represents the value ofcomfort parameter b for MPC interval τ−1, a→b indicates that powerconsumption by DER a will influence comfort parameter b, δ_(h) ^(a)represents a DER conversion efficiency factor for DER a, ΔT representsthe duration of MPC interval τ, and c_(h,τ) ^(b) represents an externalinfluence factor on comfort parameter b for MPC interval τ.

Example 8. An aggregation unit comprising: at least one processorconfigured to: receive a service collaboration request indicating aproblem in a power distribution network; responsive to receiving theservice collaboration request, determine, based on respective minimumand maximum real and reactive power values for one or more DERs in thepower distribution network under management of the aggregation unit, anoptimized operation schedule covering a model predictive control (MPC)horizon duration for the one or more DERS; determine, based on theoptimized operation schedule, an estimated flexibility range for devicesunder management by the aggregation unit; output, for use by amanagement device in the power distribution network, an indication ofthe estimated flexibility range; receive an indication of an overalloptimized operation schedule covering the MPC horizon duration;determine, based on the indication of the overall optimized operationschedule, setpoints for the one or more DERs; and cause at least one ofthe one or more DERs to modify operation based on the setpoints.

Example 9. The aggregation unit of example 8, wherein: the at least oneprocessor is further configured to determine the optimized operationschedule, determine the estimated flexibility range, and output theindication of the estimated flexibility range iteratively at a firstfrequency that represents an MPC interval the at least one processor isfurther configured to determine the setpoints for the one or more DERsand cause the at least one of the one or more DERs to modify operationiteratively at a second frequency that represents a real-time interval,and the MPC interval is larger than the real-time interval.

Example 10. The aggregation unit of example 9, wherein: determining thesetpoints for the one or more DERs is further based on a powerover-consumption threshold, an energy over-consumption threshold, andrespective minimum and maximum values of a comfort parameter for devicesunder management by the aggregation unit, the power over-consumptionthreshold represents a level of power consumption by the devices undermanagement by the aggregation unit that, if exceeded during anyreal-time interval, will incur a penalty, the energy over-consumptionthreshold represents a level of energy consumption by the devices undermanagement by the aggregation unit that, if exceeded during any MPCinterval, will incur a penalty, and the respective minimum and maximumcomfort settings represent a desired—but not required-operating range ofthe devices under management.

Example 11. The aggregation unit of any of examples 8-10, whereindetermining the estimated flexibility range comprises determining: a sumof respective minimum powers of devices under management by theaggregation unit; a sum of respective maximum power of devices undermanagement by the aggregation unit; a sum of respective minimum energyconsumption of devices under management by the aggregation unit; and asum of respective maximum energy consumption of devices under managementby the aggregation unit.

Example 12. The aggregation unit of any of examples 8-11, whereindetermining the estimated flexibility range is further based onrespective minimum and maximum values of a comfort parameter for the oneor more DERs.

Example 13. The aggregation unit of any of examples 8-12, whereindetermining the optimized operation schedule for the one or more DERSduring the MPC horizon duration comprises determining:

Min  Σ_(τ = T_(st))^(T_(en))(Σ_(a ∈ Ω_(A_(h)))p_(h, τ)^(a) + Σ_(b ∈ Ω_(B_(h)))β_(h)^(b)s_(h, τ)^(b) − s_(h)^(b, pre)₂²)

subject to: p_(h,τ) ^(a,min)≤p_(h,τ) ^(a)≤p_(h,τ) ^(a,max), q_(h,τ)^(a,min)≤q_(h,τ) ^(a)≤q_(h,τ) ^(a,max), s_(h,τ) ^(b,min)≤s_(h,τ)^(b)≤s_(h,τ) ^(b,max), and s_(h,τ) ^(b)=s_(h,τ−1) ^(b)+Σ_(a→b)δ_(h)^(a)p_(h,τ) ^(a)ΔT+c_(h,τ) ^(b)+ΔT, wherein: τ represents an MPCinterval, T_(st) and T_(en) represent a start time and an end time,respectively, of the MPC horizon, Ω_(A) _(h) represents the one or moreDERs, a represents a DER in the one or more DERs, p_(h,τ) ^(a)represents an active power of DER a for MPC interval τ, Ω_(B) _(h)represents a set of comfort parameters, b represents a comfort parameterin the set of comfort parameters, s_(h,τ) ^(b) represents a value ofcomfort parameter b for MPC interval τ, s_(h,τ) ^(b,pre) represents apreferred value of comfort parameter b for MPC interval τ, p_(h,τ)^(a,min) and p_(h,τ) ^(a,max) represent a minimum and maximum activepower of DER a, respectively, for MPC interval τ, s_(h,τ) ^(b,min) ands_(h,τ) ^(b,max) represent a desired minimum and maximum value,respectively, of comfort parameter b for MPC interval τ, s_(h,τ−1) ^(b)represents the value of comfort parameter b for MPC interval τ−1, a→bindicates that power consumption by DER a will influence comfortparameter b, δ_(h) ^(a) represents a DER conversion efficiency factorfor DER a, ΔT represents the duration of MPC interval τ, and c_(h,τ)^(b) represents an external influence factor on comfort parameter b forMPC interval τ.

Example 14. The aggregation unit of any of examples 8-13, whereindetermining the estimated flexibility range comprises determining:subject to: p_(h,τ) ^(a,min)≤p_(h,τ) ^(a)≤p_(h,τ) ^(a,max), q_(h,τ)^(a,min)≤q_(h,τ) ^(a)≤q_(h,τ) ^(a,max), s_(h,τ) ^(b,min)≤s_(h,τ)^(b)≤s_(h,τ) ^(b,max), and s_(h,τ) ^(b)=s_(h,τ−1) ^(b)+Σ_(a→b)δ_(h)^(a)p_(h,τ) ^(a)ΔT+c_(h,τ) ^(b)+ΔT, wherein: τ represents an MPCinterval, T_(st) represents a start time of the MPC horizon, e_(h,τ)^(min) and e_(h,τ) ^(max) represent a minimum and maximum energyconsumption of devices under management by the aggregation unit for MPCinterval τ, Ω_(A) _(h) represents the one or more DERs, a represents aDER in the one or more DERs, p_(h,τ) ^(a) represents an active power ofDER a for MPC interval τ, b represents a comfort parameter in a set ofcomfort parameters, s_(h,τ) ^(b) represents a value of comfort parameterb for MPC interval τ, s_(h,τ) ^(b,pre) represents a preferred value ofcomfort parameter b for MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ)^(a,max) represent a minimum and maximum active power of DER a,respectively, for MPC interval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max)represent a desired minimum and maximum value, respectively, of comfortparameter b for MPC interval τ, s_(h,τ−1) ^(b) represents the value ofcomfort parameter b for MPC interval τ−1, a→b indicates that powerconsumption by DER a will influence comfort parameter b, δ_(h) ^(a)represents a DER conversion efficiency factor for DER a, ΔT representsthe duration of MPC interval τ, and c_(h,τ) ^(b) represents an externalinfluence factor on comfort parameter b for MPC interval τ.

Example 15. A power management unit comprising: at least one processorconfigured to: receive an indication of an estimated flexibility rangeof one or more devices in a power distribution network that are undermanagement by an aggregation unit; determine, based on a linearizedrestoration model of the power distribution network that includes theindication of the estimated flexibility range, an optimized powerdistribution network reconfiguration plan and an overall optimizedoperation schedule covering the MPC horizon duration for both energyresources under management by the power management unit and one or moredistributed energy resources (DERs) under management by the aggregationunit; cause, based on the optimized power distribution networkreconfiguration plan, a reconfiguration of the power distributionnetwork; output, for use by the aggregation unit, an indication of theoverall optimized operation schedule; and cause one or more of theenergy resources under management by the power management unit to modifyoperation based on the overall optimized operation schedule.

Example 16. The power management unit of example 15, wherein: the atleast one processor is further configured to receive the indication ofthe estimated flexibility range, determine the optimized powerdistribution network reconfiguration plan and the overall optimizedoperation schedule, cause the reconfiguration of the power distributionnetwork, and output the indication of the overall optimized operationschedule iteratively at a first frequency that represents an MPCinterval, the at least one processor is further configured to cause theone or more of the energy resources under management by the powermanagement unit to modify operation iteratively at a second frequencythat represents a real-time interval, and the MPC interval is largerthan the real-time interval.

Example 17. The power management unit of any of examples 15-16, whereinthe at least one processor is further configured to: receive anindication of an optimized operation schedule for the one or moredevices, and determine the optimized power distribution networkreconfiguration plan and the overall optimized operation schedule basedfurther on the optimized operation schedule.

Example 18. The power management unit of any of examples 15-17, wherein:the at least one processor is further configured to receive anindication of an optimized operation schedule for the one or moredevices that are under management by the aggregation unit, anddetermining the overall optimized operation schedule covering the MPChorizon duration for both energy resources under management by the powermanagement unit and the one or more DERs is further based on theindication of the optimized operation schedule.

Example 19. The power management unit of any of examples 15-18, whereincausing one or more of the energy resources under management by thepower management unit to modify operation comprises: determining, basedon the overall optimized operation schedule, setpoints for the one ormore energy resources; and outputting, for use by the one or more energyresources, an indication of the setpoints.

Example 20. The power management unit of any of examples 15-19, whereindetermining the overall optimized operation schedule covering the MPChorizon duration for both energy resources under management by the powermanagement unit and the one or more DERs by determining: Min Σ_(τ=T)_(st) ^(T) ^(en) Σ_(n∈Ω) _(N) [γ_(n) ^(G)P_(n,τ) ^(G)+γ_(n) ^(S)Σ_(ϕ∈Ψ)_(n) (P_(n,τ) ^(S,ϕ)+Q_(n,τ) ^(S,ϕ))]+Σ_(τ=T) _(st) ^(T) ^(en) Σ_(g∈Ω)_(H) (γ_(h) ^(V)P_(h,τ) ^(V)+γ_(h) ^(W)P_(h,τ) ^(W)), subject to:P_(n,τ) ^(G,min)≤P_(n,τ) ^(G)≤P_(nτ) ^(G,max), Q_(n,τ) ^(G,min)≤Q_(n,τ)^(G)≤Q_(nτ) ^(G,max), P_(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(nP) _(n,τ) ^(G,ϕ), Q_(n,τ)^(G)=Σ_(ϕ∈Ψ) _(n) Q_(n,τ) ^(G,ϕ), P_(n,τ) ^(G,ϕ)−P_(n,τ) ^(D,ϕ)+P_(n,τ)^(S,ϕ)=Σ_(nm∈Ω) _(L) (P_(nm,τ) ^(ϕ)−_(mn,τ) ^(ϕ)), Q_(n,τ)^(G,ϕ)−Q_(n,τ) ^(D,ϕ)+Q_(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (Q_(nm,τ)^(ϕ)−Q_(mn,τ) ^(ϕ)), P_(n,τ) ^(D,ϕ)=P_(n,τ)^(D,fix,ϕ)+Σ_(h→(n,ϕ))P_(h,τ), Q_(n,τ) ^(D,ϕ)=Q_(n,τ)^(D,fix,ϕ)+Σ_(h→(n,ϕ))Q_(h,τ), 0≤P_(n,τ) ^(S,ϕ)≤P_(n,τ) ^(D,ϕ),0≤Q_(n,τ) ^(S,ϕ)≤Q_(n,τ) ^(D,ϕ), V_(n,τ) ^(ϕ)−V_(m,τ) ^(ϕ)=2(r_(nm)^(ϕϕ)P_(nm,τ) ^(ϕ)+x_(nm) ^(ϕϕ)Q_(nm,τ) ^(ϕ))−2ϑ₁(r_(nm) ^(ϕϕ′)P_(nm,τ)^(ϕ′)+r_(nm) ^(ϕϕ″)P_(nm,τ) ^(ϕ″)+x_(nm) ^(ϕϕ′)Q_(nm,τ) ^(ϕ′)+x_(nm)^(ϕϕ″)Q_(nm,τ) ^(ϕ″))−2ϑ₂(x_(nm) ^(ϕϕ′)P_(nm,τ) ^(ϕ′)−x_(nm)^(ϕϕ″)P_(nm,τ) ^(ϕ″)−r_(nm) ^(ϕϕ′)Q_(nm,τ) ^(ϕ′)+r_(nm) ^(ϕϕ″)Q_(nm,τ)^(ϕ″)), V_(n) ^(ϕ,min)≤V_(n,τ) ^(ϕ)≤V_(n) ^(ϕ,max), P_(h,τ)^(W)≥P_(h,τ)−p_(h,τ)*, P_(h,τ) ^(W)≥p_(h,τ)*−P_(h,τ), p_(h,τ)^(min)≤P_(h,τ)≤p_(h,τ) ^(max), e_(h,τ) ^(min)−P_(h,τ) ^(V)≤Σ_(T=T) _(st)^(τ)P_(h,T)≤e_(h,τ) ^(max)P_(h,τ) ^(V), and P_(h,τ) ^(V)≥0, wherein: τ,represents an MPC interval, T_(st) and T_(en) represent a start time andan end time, respectively, of the MPC horizon, n represents a bus in thepower distribution network, f_(N) represents a set of buses in the powerdistribution network, yn represents a cost of energy generation by theenergy resources under management by the power management unit at bus n,P_(n,τ) ^(G) and Q_(n,τ) ^(G) represent active and reactive powergeneration, respectively, by the energy resources under management bythe power management unit at bus n during MPC interval τ, γ_(n) ^(S)represents a penalty of load shed at bus n, ϕrepresents a phase inΨ_(n), Ψ_(n) represents a set of phases at bus n, P_(n,τ) ^(S,ϕ) andQ_(n,τ) ^(S,ϕ) represent active and reactive power generation,respectively, by the energy resources under management by the powermanagement unit on phase ϕ at bus n that is shed during MPC interval τ,h represents an aggregation unit in the power distribution network,Ω_(H) represents a set of aggregation units in the power distributionnetwork, γh represents a penalty of aggregation unit h violating itsestimated flexibility range, P_(n,τ) ^(V) represents a violation ofaggregation unit's h estimated flexibility range in MPC interval τ,yh^(W) represents a penalty of aggregation unit h violating itsoptimized operation schedule, P_(n,τ) ^(W) represents a deviation ofaggregation unit h from its optimized operation schedule in MPC intervalτ, P_(n,τ) ^(G,min) and P_(n,τ) ^(G,max) represent a minimum and maximumactive power generation, respectively, by the energy resources undermanagement by the power management unit at bus n during MPC interval τ,Q_(n,τ) ^(G,ϕ) and Q_(n,τ) ^(G,max) represent a minimum and maximumreactive power generation, respectively, by the energy resources undermanagement by the power management unit at bus n during MPC interval τ,P_(n,τ) ^(G,ϕ), and Q_(n,τ) ^(G,ϕ) represent active and reactive powergeneration, respectively, by the energy resources under management bythe power management unit on phase (at bus n during MPC interval τ,P_(n,τ) ^(D,ϕ), and Q_(n,τ) ^(D,ϕ) represent active and reactive loadconsumption, respectively, by phase ϕ at bus n during MPC interval τ, nmrepresents a distribution line from bus n to bus m in the powerdistribution network, Ω_(L) represents a set of distribution lines inthe power distribution network, P_(nm,τ) ^(ϕ) and Q_(nm,τ) ^(ϕ)represent the active and reactive power flow, respectively, through linenm during MPC interval τ, P_(mn,τ) ^(ϕ) and Q_(mn,τ) ^(ϕ) represent theactive and reactive power flow, respectively, through line mn during MPCinterval τ, P_(n,τ) ^(D,fix,ϕ) and Q_(n,τ) ^(D,fix,ϕ) represent activeand reactive uncontrollable load consumption, respectively, by phase (atbus n during MPC interval τ, P_(h,τ) and Q_(h,τ) represent optimizedactive and reactive loads, respectively, for aggregation unit h duringMPC interval τ, V_(n,τ) ^(ϕ) and V_(m,τ) ^(ϕ) represent a squared nodalvoltage magnitude on phase ϕ at bus n and bus m, respectively, r_(nm)^(ϕϕ) and x_(nm) ^(ϕϕ) represent the resistance and reactance,respectively, of line nm, ϕ′ and ϕ″ represent the phase that leads andlags ϕ, respectively, by 2π/3, ϑ₁ and ϑ₂ represent cos (2π/3) and sin(2π/3), respectively, V_(n) ^(ϕ,min) and V_(n) ^(ϕ,max) represent aminimum and maximum value of V_(n) ^(ϕ), respectively, P_(h,τ)*represents unit h's optimized operation schedule in MPC interval τ,P_(h,τ) ^(min) and P_(n,τ) ^(max) represent a minimum and maximum activepower, respectively, for aggregation unit h during MPC interval τ, ande_(h,τ) ^(min) and e_(h,τ) ^(max) a minimum and maximum energyconsumption, respectively, for aggregation unit h during MPC interval τ.

Example 21. A method comprising: receiving, by an aggregation unitcomprising at least one processor, a service collaboration requestindicating a problem in a power distribution network; responsive toreceiving the service collaboration request, determining, by theaggregation unit, based on respective minimum and maximum real andreactive power values for one or more DERs in the power distributionnetwork under management by the aggregation unit, an optimized operationschedule covering a model predictive control (MPC) horizon duration forthe one or more DERs; determining, by the aggregation unit, based on theoptimized operation schedule, an estimated flexibility range for devicesunder management by the aggregation unit; determining, by a powermanagement unit comprising at least one processor, based on a linearizedrestoration model of the three-phase unbalanced power distributionnetwork that includes an indication of the estimated flexibility range,an optimized power distribution network reconfiguration plan and anoverall optimized operation schedule covering the MPC horizon durationfor both energy resources under management by the power management unitand the one or more DERs; causing, by the power management unit, basedon the optimized power distribution network reconfiguration plan, areconfiguration of the power distribution network; causing, by the powermanagement unit, one or more of the energy resources under management bythe power management unit to modify operation based on the overalloptimized operation schedule, determining, by the aggregation unit,based on the indication of the overall optimized operation schedule,setpoints for the one or more DERs; and causing, by the aggregationunit, at least one of the one or more DERs to modify operation based onthe setpoints.

Example 22: The method of example 21, further comprising operations toperform any of the techniques of examples 2-7

Example 23: A method comprising: receiving, by an aggregation unitcomprising at least one processor, a service collaboration requestindicating a problem in a power distribution network; responsive toreceiving the service collaboration request, determining, by theaggregation unit, based on respective minimum and maximum real andreactive power values for one or more DERs in the power distributionnetwork under management of the aggregation unit, an optimized operationschedule covering a model predictive control (MPC) horizon duration forthe one or more DERS; determining, by the aggregation unit, based on theoptimized operation schedule, an estimated flexibility range for devicesunder management by the aggregation unit; outputting, by the aggregationunit, for use by a management device in the power distribution network,an indication of the estimated flexibility range; receiving, by theaggregation unit, an indication of an overall optimized operationschedule covering the MPC horizon duration; determining, by theaggregation unit, based on the indication of the overall optimizedoperation schedule, setpoints for the one or more DERs; and causing, bythe aggregation unit, at least one of the one or more DERs to modifyoperation based on the setpoints.

Example 24: The method of example 23, further comprising operations toperform any of the techniques of examples 9-14.

Example 25: A method comprising: receiving, by a power management unitcomprising at least one processor, an indication of an estimatedflexibility range of one or more devices in a power distribution networkthat are under management by an aggregation unit; determining, by thepower management unit, based on a linearized restoration model of thepower distribution network that includes the indication of the estimatedflexibility range, an optimized power distribution networkreconfiguration plan and an overall optimized operation schedulecovering the MPC horizon duration for both energy resources undermanagement by the power management unit and one or more distributedenergy resources (DERs) under management by the aggregation unit;causing, by the power management unit, based on the optimized powerdistribution network reconfiguration plan, a reconfiguration of thepower distribution network; outputting, by the power management unit,for use by the aggregation unit, an indication of the overall optimizedoperation schedule; and causing, by the power management unit, one ormore of the energy resources under management by the power managementunit to modify operation based on the overall optimized operationschedule.

Example 26: The method of example 25, further comprising operations toperform any of the techniques of examples 16-20.

Example 27: A non-transitory computer readable medium encoded withinstructions that, when executed, cause at least one processor toperform any of the techniques of examples 1-20. In one or more examples,the techniques described herein may be implemented in hardware,software, firmware, or any combination thereof. If implemented insoftware, the functions may be stored on or transmitted over, as one ormore instructions or code, a computer-readable medium and executed by ahardware-based processing unit. Computer-readable media may includecomputer-readable storage media, which corresponds to a tangible mediumsuch as data storage media, or communication media, which includes anymedium that facilitates transfer of a computer program from one place toanother, e.g., according to a communication protocol. In this manner,computer-readable media generally may correspond to (1) tangiblecomputer-readable storage media, which is non-transitory or (2) acommunication medium such as a signal or carrier wave. Data storagemedia may be any available media that can be accessed by one or morecomputers or one or more processors to retrieve instructions, codeand/or data structures for implementation of the techniques described inthis disclosure. A computer program product may include acomputer-readable storage medium.

By way of example, and not limitation, such computer-readable storagemedia can comprise RAM, ROM, EEPROM, CD-ROM or other optical diskstorage, magnetic disk storage, or other magnetic storage devices, flashmemory, or any other medium that can be used to store desired programcode in the form of instructions or data structures and that can beaccessed by a computer. Also, any connection is properly termed acomputer-readable medium. For example, if instructions are transmittedfrom a website, server, or other remote source using a coaxial cable,fiber optic cable, twisted pair, digital subscriber line (DSL), orwireless technologies such as infrared, radio, and microwave, then thecoaxial cable, fiber optic cable, twisted pair, DSL, or wirelesstechnologies such as infrared, radio, and microwave are included in thedefinition of medium. It should be understood, however, thatcomputer-readable storage media and data storage media do not includeconnections, carrier waves, signals, or other transient media, but areinstead directed to non-transient, tangible storage media. Disk anddisc, as used herein, includes compact disc (CD), laser disc, opticaldisc, digital versatile disc (DVD), floppy disk and Blu-ray disc, wheredisks usually reproduce data magnetically, while discs reproduce dataoptically with lasers. Combinations of the above should also be includedwithin the scope of computer-readable media.

Instructions may be executed by one or more processors, such as one ormore digital signal processors (DSPs), general purpose microprocessors,application specific integrated circuits (ASICs), field programmablelogic arrays (FPGAs), or other equivalent integrated or discrete logiccircuitry. Accordingly, the term “processor,” as used herein may referto any of the foregoing structure or any other structure suitable forimplementation of the techniques described herein. In addition, in someaspects, the functionality described herein may be provided withindedicated hardware and/or software modules. Also, the techniques couldbe fully implemented in one or more circuits or logic elements.

The techniques of this disclosure may be implemented in a wide varietyof devices or apparatuses, including a wireless handset, an integratedcircuit (IC) or a set of ICs (e.g., a chip set). Various components,modules, or units are described in this disclosure to emphasizefunctional aspects of devices configured to perform the disclosedtechniques, but do not necessarily require realization by differenthardware units. Rather, as described above, various units may becombined in a hardware unit or provided by a collection ofinter-operative hardware units, including one or more processors asdescribed above, in conjunction with suitable software and/or firmware.

The foregoing disclosure includes various examples set forth merely asillustration. The disclosed examples are not intended to be limiting.Modifications incorporating the spirit and substance of the describedexamples may occur to persons skilled in the art. These and otherexamples are within the scope of this disclosure and the followingclaims.

What is claimed is:
 1. A system comprising: an aggregation unitcomprising at least one processor, the aggregation unit being configuredto: receive a service collaboration request indicating a problem in apower distribution network; responsive to receiving the servicecollaboration request, determine, based on respective minimum andmaximum real and reactive power values for one or more DERs in the powerdistribution network under management by the aggregation unit, anoptimized operation schedule covering a model predictive control (MPC)horizon duration for the one or more DERs; determine, based on theoptimized operation schedule, an estimated flexibility range for devicesunder management by the aggregation unit; and output an indication ofthe estimated flexibility range; a power management unit comprising atleast one processor, the power management unit being configured to:receive the indication of the estimated flexibility range; determine,based on a linearized restoration model of the three-phase unbalancedpower distribution network that includes the indication of the estimatedflexibility range, an optimized power distribution networkreconfiguration plan and an overall optimized operation schedulecovering the MPC horizon duration for both energy resources undermanagement by the power management unit and the one or more DERs; cause,based on the optimized power distribution network reconfiguration plan,a reconfiguration of the power distribution network; output anindication of the overall optimized operation schedule; and cause one ormore of the energy resources under management by the power managementunit to modify operation based on the overall optimized operationschedule, wherein the aggregation unit is further configured to: receivethe indication of the overall optimized operation schedule; determine,based on the indication of the overall optimized operation schedule,setpoints for the one or more DERs; and cause at least one of the one ormore DERs to modify operation based on the setpoints.
 2. The system ofclaim 1, wherein: the aggregation unit is further configured todetermine the optimized operation schedule, determine the estimatedflexibility range, and output the indication of the estimatedflexibility range iteratively at a first frequency that represents anMPC interval, the power management unit is further configured to receivethe indication of the estimated flexibility range, determine theoptimized power distribution network reconfiguration plan and theoverall optimized operation schedule, cause the reconfiguration of thepower distribution network, and output the indication of the overalloptimized operation schedule iteratively at the first frequency, theaggregation unit is further configured to determine the setpoints forthe one or more DERs and cause the at least one of the one or more DERsto modify operation iteratively at a second frequency that represents areal-time interval, the power management unit is further configured tocause the one or more of the energy resources under management by thepower management unit to modify operation iteratively at the secondfrequency, and the MPC interval is larger than the real-time interval.3. The system of claim 1, wherein: the aggregation unit is furtherconfigured to output an indication of the optimized operation schedule,and the power management unit is configured to determine the optimizedpower distribution network reconfiguration plan and the overalloptimized operation schedule based further on the optimized operationschedule.
 4. The system of claim 1, wherein the aggregation unit isconfigured to cause at least one of the one or more DERs to modifyoperation based on the setpoints by outputting, for use by the at leastone of the one or more DERs, an indication of the setpoints.
 5. Thesystem of claim 1, wherein the power management unit is configured todetermine the overall optimized operation schedule covering the MPChorizon duration for both energy resources under management by the powermanagement unit and the one or more DERs by determining:Min Σ_(τ=T) _(st) ^(T) ^(en) Σ_(n∈Ω) _(N) [γ_(n) ^(G) P _(n,τ)^(G)+γ_(n) ^(S)Σ_(ϕ∈Ψ) _(n) (P _(n,τ) ^(S,ϕ) +Q _(n,τ) ^(S,ϕ))]+Σ_(τ=T)_(st) ^(T) ^(en) Σ_(g∈Ω) _(H) (γ_(h) ^(V) P _(h,τ) ^(V)+γ_(h) ^(W) P_(h,τ) ^(W)),subject to:P _(n,τ) ^(G,min) ≤P _(n,τ) ^(G) ≤P _(nτ) ^(G,max),Q _(n,τ) ^(G,min) ≤Q _(n,τ) ^(G) ≤Q _(nτ) ^(G,max),P _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) P _(n,τ) ^(G,ϕ) ,Q _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) Q_(n,τ) ^(G,ϕ),P _(n,τ) ^(G,ϕ) −P _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (P_(nm,τ) ^(ϕ)−_(mn,τ) ^(ϕ),Q _(n,τ) ^(G,ϕ) −Q _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (Q_(nm,τ) ^(ϕ) −Q _(mn,τ) ^(ϕ)),P _(n,τ) ^(D,ϕ) =P _(n,τ) ^(D,fix,ϕ)+Σ_(h→) P _(h,τ),Q _(n,τ) ^(D,ϕ) =Q _(n,τ) ^(D,fix,ϕ)+Σ_(h→) Q _(h,τ)  (16)0≤P _(n,τ) ^(S,ϕ) ≤P _(n,τ) ^(D,ϕ),0≤Q _(n,τ) ^(S,ϕ) ≤Q _(n,τ) ^(D,ϕ),V _(n,τ) ^(ϕ) −V _(m,τ) ^(ϕ)=2(r _(nm) ^(ϕϕ) P _(nm,τ) ^(ϕ) +x _(nm)^(ϕϕ) Q _(nm,τ) ^(ϕ))−2ϑ₁(r _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″)P _(nm,τ) ^(ϕ″) +x _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +x _(nm) ^(ϕϕ″) Q_(nm,τ) ^(ϕ″))−2ϑ₂(x _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) −x _(nm) ^(ϕϕ″) P_(nm,τ) ^(ϕ″) −r _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″) Q _(nm,τ)^(ϕ″)),V _(n) ^(ϕ,min) ≤V _(n,τ) ^(ϕ) ≤V _(n) ^(ϕ,max),P _(h,τ) ^(W) ≥P _(h,τ) −p _(h,τ)*,P _(h,τ) ^(W) ≥p _(h,τ) *−P _(h,τ),p _(h,τ) ^(min) ≤P _(h,τ) ≤p _(h,τ) ^(max),e _(h,τ) ^(min) −P _(h,τ) ^(V)≤Σ_(T=T) _(st) ^(τ) P _(h,T) ≤e _(h,τ)^(max) P _(h,τ) ^(V),P _(h,τ) ^(V)≥0, wherein: τ represents an MPC interval, T_(st) andT_(en) represent a start time and an end time, respectively, of the MPChorizon, n represents a bus in the power distribution network, Ω_(N)represents a set of buses in the power distribution network, γ_(n) ^(G)represents a cost of energy generation by the energy resources undermanagement by the power management unit at bus n, P_(n,τ) ^(G) andQ_(n,τ) ^(G) represent active and reactive power generation,respectively, by the energy resources under management by the powermanagement unit at bus n during MPC interval τ, γ_(n) ^(S) represents apenalty of load shed at bus n, ϕ represents a phase in Ψ_(n), Ψ_(n)represents a set of phases at bus n, P_(n,τ) ^(S,ϕ) and Q_(n,τ) ^(S,ϕ)represent active and reactive power generation, respectively, by theenergy resources under management by the power management unit on phaseϕ at bus n that is shed during MPC interval τ, h represents anaggregation unit in the power distribution network, Ω_(H) represents aset of aggregation units in the power distribution network, γ_(h) ^(V)represents a penalty of aggregation unit h violating its estimatedflexibility range, P_(h,τ) ^(W) represents a violation of aggregationunit's h estimated flexibility range in MPC interval τ, γ_(h) ^(W)represents a penalty of aggregation unit h violating its optimizedoperation schedule, P_(h,τ) ^(W) represents a deviation of aggregationunit h from its optimized operation schedule in MPC interval τ, P_(n,τ)^(G,min) and P_(n,τ) ^(G,max) represent a minimum and maximum activepower generation, respectively, by the energy resources under managementby the power management unit at bus n during MPC interval τ, Q_(n,τ)^(G,min) and Q_(n,τ) ^(G,max) represent a minimum and maximum reactivepower generation, respectively, by the energy resources under managementby the power management unit at bus n during MPC interval τ, P_(n,τ)^(G,ϕ) and Q_(n,τ) ^(G,ϕ) represent active and reactive powergeneration, respectively, by the energy resources under management bythe power management unit on phase ϕ at bus n during MPC interval τ, τ,P_(n,τ) ^(D,ϕ), and Q_(n,τ) ^(D,ϕ) represent active and reactive loadconsumption, respectively, by phase ϕ at bus n during MPC interval τ, nmrepresents a distribution line from bus n to bus m in the powerdistribution network, Ω_(L) represents a set of distribution lines inthe power distribution network, P_(nm,τ) ^(ϕ) and Q_(nm,τ) ^(ϕ)represent the active and reactive power flow, respectively, through linenm during MPC interval τ, P_(mn,τ) ^(ϕ) and Q_(mn,τ) ^(ϕ) represent theactive and reactive power flow, respectively, through line mn during MPCinterval τ, P_(n,τ) ^(D,fix,ϕ) and Q_(n,τ) ^(D,fix,ϕ) represent activeand reactive uncontrollable load consumption, respectively, by phase ϕat bus n during MPC interval τ, τ, P_(h,τ) and Q_(h,τ) representoptimized active and reactive loads, respectively, for aggregation unith during MPC interval τ, V_(n,τ) ^(ϕ) and V_(m,τ) ^(ϕ) represent asquared nodal voltage magnitude on phase ϕ at bus n and bus m,respectively, r_(nm) ^(ϕϕ) and x_(nm) ^(ϕϕ) represent the resistance andreactance, respectively, of line nm, ϕ′ and ϕ″ represent the phase thatleads and lags ϕ, respectively, by 2π/3, ϑ₁ and ϑ₂ represent cos (2π/3)and sin (2π/3), respectively, V_(n) ^(ϕ,min) and V_(n) ^(ϕ,max)represent a minimum and maximum value of V_(n) ^(ϕ), respectively,P_(h,τ)* represents unit h's optimized operation schedule in MPCinterval τ, P_(h,τ) ^(min) and P_(n,τ) ^(max) represent a minimum andmaximum active power, respectively, for aggregation unit h during MPCinterval τ, and e_(h,τ) ^(min) and e_(h,τ) ^(max) a minimum and maximumenergy consumption, respectively, for aggregation unit h during MPCinterval τ.
 6. The system of claim 1, wherein the aggregation unit isconfigured to determine the optimized operation schedule covering theMPC horizon for the one or more DERS by determining:p _(h,τ) ^(a,min) ≤p _(h,τ) ^(a) ≤p _(h,τ) ^(a,max),q _(h,τ) ^(a,min) ≤q _(h,τ) ^(a) ≤q _(h,τ) ^(a,max),s _(h,τ) ^(b,min) ≤s _(h,τ) ^(b) ≤s _(h,τ) ^(b,max), ands _(h,τ) ^(b) =s _(h,τ−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,τ) ^(a) ΔT+c_(h,τ) ^(b) +ΔT, wherein: τ represents an MPC interval, T_(st) andT_(en) represent a start time and an end time, respectively, of the MPChorizon, Ω_(A) _(h) represents the one or more DERs, a represents a DERin the one or more DERs, p_(h,τ) ^(a) represents an active power of DERa for MPC interval τ, Ω_(B) _(h) represents a set of comfort parameters,b represents a comfort parameter in the set of comfort parameters,s_(h,τ) ^(b) represents a value of comfort parameter b for MPC intervalτ, s_(h,τ) ^(b,pre) represents a preferred value of comfort parameter bfor MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ) ^(a,max) represent aminimum and maximum active power of DER a, respectively, for MPCinterval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max) represent a desiredminimum and maximum value, respectively, of comfort parameter b for MPCinterval τ, s_(h,τ−1) ^(b) represents the value of comfort parameter bfor MPC interval τ−1, a→b indicates that power consumption by DER a willinfluence comfort parameter b, δ_(h) ^(a) represents a DER conversionefficiency factor for DER a, ΔT represents the duration of MPC intervalτ, and c_(h,τ) ^(b) represents an external influence factor on comfortparameter b for MPC interval τ.
 7. The system of claim 1, wherein theaggregation unit is configured to determine the estimated flexibilityrange by determining:p _(h,τ) ^(a,min) ≤p _(h,τ) ^(a) ≤p _(h,τ) ^(a,max),q _(h,τ) ^(a,min) ≤q _(h,τ) ^(a) ≤q _(h,τ) ^(a,max),s _(h,τ) ^(b,min) ≤s _(h,τ) ^(b) ≤s _(h,τ) ^(b,max), ands _(h,τ) ^(b) =s _(h,τ−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,τ) ^(a) ΔT+c_(h,τ) ^(b) +ΔT, wherein: T represents an MPC interval, T_(st)represents a start time of the MPC horizon, e_(h,τ) ^(min) and e_(h,τ)^(max) represent a minimum and maximum energy consumption of devicesunder management by the aggregation unit for MPC interval τ, Ω_(A) _(h)represents the one or more DERs, a represents a DER in the one or moreDERs, p_(h,τ) ^(a) represents an active power of DER a for MPC intervalτ, b represents a comfort parameter in a set of comfort parameters,s_(h,τ) ^(b) represents a value of comfort parameter b for MPC intervalτ, s_(h,τ) ^(b,pre) represents a referred value of comfort parameter bfor MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ) ^(a,max) represent aminimum and maximum active power of DER a, respectively, for MPCinterval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max) represent a desiredminimum and maximum value, respectively, of comfort parameter b for MPCinterval τ, s_(h,τ−1) ^(b) represents the value of comfort parameter bfor MPC interval τ−1, a→b indicates that power consumption by DER a willinfluence comfort parameter b, δ_(h) ^(a) represents a DER conversionefficiency factor for DER a, ΔT represents the duration of MPC intervalτ, and c_(h,τ) ^(b) represents an external influence factor on comfortparameter b for MPC interval τ.
 8. An aggregation unit comprising: atleast one processor configured to: receive a service collaborationrequest indicating a problem in a power distribution network; responsiveto receiving the service collaboration request, determine, based onrespective minimum and maximum real and reactive power values for one ormore DERs in the power distribution network under management of theaggregation unit, an optimized operation schedule covering a modelpredictive control (MPC) horizon duration for the one or more DERS;determine, based on the optimized operation schedule, an estimatedflexibility range for devices under management by the aggregation unit;output, for use by a management device in the power distributionnetwork, an indication of the estimated flexibility range; receive anindication of an overall optimized operation schedule covering the MPChorizon duration; determine, based on the indication of the overalloptimized operation schedule, setpoints for the one or more DERs; andcause at least one of the one or more DERs to modify operation based onthe setpoints.
 9. The aggregation unit of claim 8, wherein: the at leastone processor is further configured to determine the optimized operationschedule, determine the estimated flexibility range, and output theindication of the estimated flexibility range iteratively at a firstfrequency that represents an MPC interval, the at least one processor isfurther configured to determine the setpoints for the one or more DERsand cause the at least one of the one or more DERs to modify operationiteratively at a second frequency that represents a real-time interval,and the MPC interval is larger than the real-time interval.
 10. Theaggregation unit of claim 9, wherein: determining the setpoints for theone or more DERs is further based on a power over-consumption threshold,an energy over-consumption threshold, and respective minimum and maximumvalues of a comfort parameter for devices under management by theaggregation unit, the power over-consumption threshold represents alevel of power consumption by the devices under management by theaggregation unit that, if exceeded during any real-time interval, willincur a penalty, the energy over-consumption threshold represents alevel of energy consumption by the devices under management by theaggregation unit that, if exceeded during any MPC interval, will incur apenalty, and the respective minimum and maximum comfort settingsrepresent a desired—but not required-operating range of the devicesunder management.
 11. The aggregation unit of claim 8, whereindetermining the estimated flexibility range comprises determining: a sumof respective minimum powers of devices under management by theaggregation unit; a sum of respective maximum power of devices undermanagement by the aggregation unit; a sum of respective minimum energyconsumption of devices under management by the aggregation unit; and asum of respective maximum energy consumption of devices under managementby the aggregation unit.
 12. The aggregation unit of claim 8, whereindetermining the estimated flexibility range is further based onrespective minimum and maximum values of a comfort parameter for the oneor more DERs.
 13. The aggregation unit of claim 8, wherein determiningthe optimized operation schedule for the one or more DERS during the MPChorizon duration comprises determining:p _(h,τ) ^(a,min) ≤p _(h,τ) ^(a) ≤p _(h,τ) ^(a,max),q _(h,τ) ^(a,min) ≤q _(h,τ) ^(a) ≤q _(h,τ) ^(a,max),s _(h,τ) ^(b,min) ≤s _(h,τ) ^(b) ≤s _(h,τ) ^(b,max), ands _(h,τ) ^(b) =s _(h,τ−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,τ) ^(a) ΔT+c_(h,τ) ^(b) +ΔT, wherein: τ represents an MPC interval, T_(st) andT_(en) represent a start time and an end time, respectively, of the MPChorizon, Ω_(A) _(h) represents the one or more DERs, a represents a DERin the one or more DERs, p_(h,τ) ^(a) represents an active power of DERa for MPC interval τ, Ω_(B) _(h) represents a set of comfort parameters,b represents a comfort parameter in the set of comfort parameters,s_(h,τ) ^(b) represents a value of comfort parameter b for MPC intervalτ, s_(h,τ) ^(b,pre) represents a preferred value of comfort parameter bfor MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ) ^(a,max) represent aminimum and maximum active power of DER a, respectively, for MPCinterval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max) represent a desiredminimum and maximum value, respectively, of comfort parameter b for MPCinterval τ, s_(h,τ−1) ^(b) represents the value of comfort parameter bfor MPC interval τ−1, a→b indicates that power consumption by DER a willinfluence comfort parameter b, δ_(h) ^(a) represents a DER conversionefficiency factor for DER a, ΔT represents the duration of MPC intervalτ, and c_(h,τ) ^(b) represents an external influence factor on comfortparameter b for MPC interval τ.
 14. The aggregation unit of claim 8,wherein determining the estimated flexibility range comprisesdetermining:p _(h,τ) ^(a,min) ≤p _(h,τ) ^(a) ≤p _(h,τ) ^(a,max),q _(h,τ) ^(a,min) ≤q _(h,τ) ^(a) ≤q _(h,τ) ^(a,max),s _(h,τ) ^(b,min) ≤s _(h,τ) ^(b) ≤s _(h,τ) ^(b,max), ands _(h,τ) ^(b) =s _(h,τ−1) ^(b)+Σ_(a→b)δ_(h) ^(a) p _(h,τ) ^(a) ΔT+c_(h,τ) ^(b) +ΔT, wherein: T represents an MPC interval, T_(st)represents a start time of the MPC horizon, e_(h,τ) ^(min) and e_(h,τ)^(max) represent a minimum and maximum energy consumption of devicesunder management by the aggregation unit for MPC interval τ, Ω_(A) _(h)represents the one or more DERs, a represents a DER in the one or moreDERs, p_(h,τ) ^(a) represents an active power of DER a for MPC intervalτ, b represents a comfort parameter in a set of comfort parameters,s_(h,τ) ^(b) represents a value of comfort parameter b for MPC intervalτ, s_(h,τ) ^(b,pre) represents a referred value of comfort parameter bfor MPC interval τ, p_(h,τ) ^(a,min) and p_(h,τ) ^(a,max) represent aminimum and maximum active power of DER a, respectively, for MPCinterval τ, s_(h,τ) ^(b,min) and s_(h,τ) ^(b,max) represent a desiredminimum and maximum value, respectively, of comfort parameter b for MPCinterval τ, s_(h,τ−1) ^(b) represents the value of comfort parameter bfor MPC interval τ−1, a→b indicates that power consumption by DER a willinfluence comfort parameter b, δ_(h) ^(a) represents a DER conversionefficiency factor for DER a, ΔT represents the duration of MPC intervalτ, and c_(h,τ) ^(b) represents an external influence factor on comfortparameter b for MPC interval τ.
 15. A power management unit comprising:at least one processor configured to: receive an indication of anestimated flexibility range of one or more devices in a powerdistribution network that are under management by an aggregation unit;determine, based on a linearized restoration model of the powerdistribution network that includes the indication of the estimatedflexibility range, an optimized power distribution networkreconfiguration plan and an overall optimized operation schedulecovering the MPC horizon duration for both energy resources undermanagement by the power management unit and one or more distributedenergy resources (DERs) under management by the aggregation unit; cause,based on the optimized power distribution network reconfiguration plan,a reconfiguration of the power distribution network; output, for use bythe aggregation unit, an indication of the overall optimized operationschedule; and cause one or more of the energy resources under managementby the power management unit to modify operation based on the overalloptimized operation schedule.
 16. The power management unit of claim 15,wherein: the at least one processor is further configured to receive theindication of the estimated flexibility range, determine the optimizedpower distribution network reconfiguration plan and the overalloptimized operation schedule, cause the reconfiguration of the powerdistribution network, and output the indication of the overall optimizedoperation schedule iteratively at a first frequency that represents anMPC interval, the at least one processor is further configured to causethe one or more of the energy resources under management by the powermanagement unit to modify operation iteratively at a second frequencythat represents a real-time interval, and the MPC interval is largerthan the real-time interval.
 17. The power management unit of claim 15,wherein the at least one processor is further configured to: receive anindication of an optimized operation schedule for the one or moredevices, and determine the optimized power distribution networkreconfiguration plan and the overall optimized operation schedule basedfurther on the optimized operation schedule.
 18. The power managementunit of claim 15, wherein: the at least one processor is furtherconfigured to receive an indication of an optimized operation schedulefor the one or more devices that are under management by the aggregationunit, and determining the overall optimized operation schedule coveringthe MPC horizon duration for both energy resources under management bythe power management unit and the one or more DERs is further based onthe indication of the optimized operation schedule.
 19. The powermanagement unit of claim 15, wherein causing one or more of the energyresources under management by the power management unit to modifyoperation comprises: determining, based on the overall optimizedoperation schedule, setpoints for the one or more energy resources; andoutputting, for use by the one or more energy resources, an indicationof the setpoints.
 20. The power management unit of claim 15, whereindetermining the overall optimized operation schedule covering the MPChorizon duration for both energy resources under management by the powermanagement unit and the one or more DERs comprises determining:Min Σ_(τ=T) _(st) ^(T) ^(en) Σ_(n∈Ω) _(N) [γ_(n) ^(G) P _(n,τ)^(G)+γ_(n) ^(S)Σ_(ϕ∈Ψ) _(n) (P _(n,τ) ^(S,ϕ) +Q _(n,τ) ^(S,ϕ))]+Σ_(τ=T)_(st) ^(T) ^(en) Σ_(g∈Ω) _(H) (γ_(h) ^(V) P _(h,τ) ^(V)+γ_(h) ^(W) P_(h,τ) ^(W)), subject to:P _(n,τ) ^(G,min) ≤P _(n,τ) ^(G) ≤P _(nτ) ^(G,max),Q _(n,τ) ^(G,min) ≤Q _(n,τ) ^(G) ≤Q _(nτ) ^(G,max),P _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) P _(n,τ) ^(G,ϕ) ,Q _(n,τ) ^(G)=Σ_(ϕ∈Ψ) _(n) Q_(n,τ) ^(G,ϕ),P _(n,τ) ^(G,ϕ) −P _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (P_(nm,τ) ^(ϕ)−_(mn,τ) ^(ϕ),Q _(n,τ) ^(G,ϕ) −Q _(n,τ) ^(D,ϕ) +P _(n,τ) ^(S,ϕ)=Σ_(nm∈Ω) _(L) (Q_(nm,τ) ^(ϕ) −Q _(mn,τ) ^(ϕ)),P _(n,τ) ^(D,ϕ) =P _(n,τ) ^(D,fix,ϕ)+Σ_(h→) P _(h,τ),Q _(n,τ) ^(D,ϕ) =Q _(n,τ) ^(D,fix,ϕ)+Σ_(h→) Q _(h,τ)  (16)0≤P _(n,τ) ^(S,ϕ) ≤P _(n,τ) ^(D,ϕ),0≤Q _(n,τ) ^(S,ϕ) ≤Q _(n,τ) ^(D,ϕ),V _(n,τ) ^(ϕ) −V _(m,τ) ^(ϕ)=2(r _(nm) ^(ϕϕ) P _(nm,τ) ^(ϕ) +x _(nm)^(ϕϕ) Q _(nm,τ) ^(ϕ))−2ϑ₁(r _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″)P _(nm,τ) ^(ϕ″) +x _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +x _(nm) ^(ϕϕ″) Q_(nm,τ) ^(ϕ″))−2ϑ₂(x _(nm) ^(ϕϕ′) P _(nm,τ) ^(ϕ′) −x _(nm) ^(ϕϕ″) P_(nm,τ) ^(ϕ″) −r _(nm) ^(ϕϕ′) Q _(nm,τ) ^(ϕ′) +r _(nm) ^(ϕϕ″) Q _(nm,τ)^(ϕ″)),V _(n) ^(ϕ,min) ≤V _(n,τ) ^(ϕ) ≤V _(n) ^(ϕ,max),P _(h,τ) ^(W) ≥P _(h,τ) −p _(h,τ)*,P _(h,τ) ^(W) ≥p _(h,τ) *−P _(h,τ),p _(h,τ) ^(min) ≤P _(h,τ) ≤p _(h,τ) ^(max),e _(h,τ) ^(min) −P _(h,τ) ^(V)≤Σ_(T=T) _(st) ^(τ) P _(h,T) ≤e _(h,τ)^(max) P _(h,τ) ^(V),P _(h,τ) ^(V)≥0, wherein: τ represents an MPC interval, T_(st) andT_(en) represent a start time and an end time, respectively, of the MPChorizon, n represents a bus in the power distribution network, Ω_(N)represents a set of buses in the power distribution network, γ_(n) ^(G)represents a cost of energy generation by the energy resources undermanagement by the power management unit at bus n, P_(n,τ) ^(G) andQ_(n,τ) ^(G) represent active and reactive power generation,respectively, by the energy resources under management by the powermanagement unit at bus n during MPC interval τ, γ_(n) ^(S) represents apenalty of load shed at bus n, ϕ represents a phase in Ψ_(n), Ψ_(n)represents a set of phases at bus n, P_(n,τ) ^(S,ϕ) and Q_(n,τ) ^(S,ϕ)represent active and reactive power generation, respectively, by theenergy resources under management by the power management unit on phaseϕ at bus n that is shed during MPC interval τ, h represents anaggregation unit in the power distribution network, Ω_(H) represents aset of aggregation units in the power distribution network, γ_(h) ^(V)represents a penalty of aggregation unit h violating its estimatedflexibility range, P_(h,τ) ^(W) represents a violation of aggregationunit's h estimated flexibility range in MPC interval τ, γ_(h) ^(W)represents a penalty of aggregation unit h violating its optimizedoperation schedule, P_(h,τ) ^(W) represents a deviation of aggregationunit h from its optimized operation schedule in MPC interval τ, P_(n,τ)^(G,min) and P_(n,τ) ^(G,max) represent a minimum and maximum activepower generation, respectively, by the energy resources under managementby the power management unit at bus n during MPC interval τ, Q_(n,τ)^(G,min) and Q_(n,τ) ^(G,max) represent a minimum and maximum reactivepower generation, respectively, by the energy resources under managementby the power management unit at bus n during MPC interval τ, P_(n,τ)^(G,ϕ) and Q_(n,τ) ^(G,ϕ) represent active and reactive powergeneration, respectively, by the energy resources under management bythe power management unit on phase ϕ at bus n during MPC interval τ, τ,P_(n,τ) ^(D,ϕ), and Q_(n,τ) ^(D,ϕ) represent active and reactive loadconsumption, respectively, by phase ϕ at bus n during MPC interval τ, nmrepresents a distribution line from bus n to bus m in the powerdistribution network, Ω_(L) represents a set of distribution lines inthe power distribution network, P_(nm,τ) ^(ϕ) and Q_(nm,τ) ^(ϕ)represent the active and reactive power flow, respectively, through linenm during MPC interval τ, P_(mn,τ) ^(ϕ) and Q_(mn,τ) ^(ϕ) represent theactive and reactive power flow, respectively, through line mn during MPCinterval τ, P_(n,τ) ^(D,fix,ϕ) and Q_(n,τ) ^(D,fix,ϕ) represent activeand reactive uncontrollable load consumption, respectively, by phase ϕat bus n during MPC interval τ, τ, P_(h,τ) and Q_(h,τ) representoptimized active and reactive loads, respectively, for aggregation unith during MPC interval τ, V_(n,τ) ^(ϕ) and V_(m,τ) ^(ϕ) represent asquared nodal voltage magnitude on phase ϕ at bus n and bus m,respectively, r_(nm) ^(ϕϕ) and x_(nm) ^(ϕϕ) represent the resistance andreactance, respectively, of line nm, ϕ′ and ϕ″ represent the phase thatleads and lags ϕ, respectively, by 2π/3, ϑ₁ and ϑ₂ represent cos (2π/3)and sin (2π/3), respectively, V_(n) ^(ϕ,min) and V_(n) ^(ϕ,max)represent a minimum and maximum value of V_(n) ^(ϕ), respectively,P_(h,τ)* represents unit h's optimized operation schedule in MPCinterval τ, P_(h,τ) ^(min) and P_(n,τ) ^(max) represent a minimum andmaximum active power, respectively, for aggregation unit h during MPCinterval τ, and e_(h,τ) ^(min) and e_(h,τ) ^(max) a minimum and maximumenergy consumption, respectively, for aggregation unit h during MPCinterval τ.